$\lambda \in *\mathbb{C}_c^#$ is in the resolvent set, $\rho(T)$, if $\lambda I - T$ is a bijection of $D(T)$ onto $H$ with a finitely or hyper finitely bounded inverse. If $\lambda \in \rho(T)$, $R_\lambda(T) = (\lambda I - T)^{-1}$ is called the resolvent of $T$ at $\lambda$.

The definitions of spectrum, point spectrum, and residual spectrum are the same for unbounded operators as they are for bounded operators. We will sometimes refer to the spectrum of nonclosed, but closable operators. In this case we always mean the spectrum of the closure.

### 3. Symmetric and self-#-adjoint operators: the basic criterion for self-#-adjointness.

**Definition 3.1.** A #-densely defined operator $T$ on a non-Archimedean Hilbert space is called symmetric (or Hermitian) if $T \subset T^*$, that is, if $D(T) \subset D(T^*)$ and $T\varphi = T^*\varphi$ for all $\varphi \in D(T)$.

Equivalently, $T$ is symmetric if and only if $(T\varphi, \psi) = (\varphi, T\psi)$ for all $\varphi, \psi \in D(T)$.

**Definition 3.2.** $T$ is called self-adjoint if $T = T^*$, that is, if and only if $T$ is symmetric and $D(T) = D(T^*)$.

A symmetric operator is always #-closable, since $D(T^*) \supset D(T)$ is #-dense in $H$. If $T$ is symmetric, $T^*$ is a closed extension of $T$ so the smallest #-closed extension $T^{**}$ of $T$ must be contained in $T^*$. Thus for symmetric operators, we have $T \subset T^{**} \subset T^*$. For #-closed symmetric operators, $T = T^{**} \subset T^*$ and, for self-adjoint operators, $T = T^{**} = T^*$.

From this one can easily see that a #-closed symmetric operator $T$ is self-adjoint if and only if $T^*$ is symmetric.

The distinction between #-closed symmetric operators and self-adjoint operators is very

important. It is only for self-adjoint operators that the spectral theorem holds and it is only self-adjoint operators that may be #-exponentiated to give the one-parameter unitary groups which give the dynamics in

QFT. Chapter X is mainly devoted to studying methods for proving that operators are self-adjoint. We content ourselves here with proving the basic criterion for selfadjointness.

First, we introduce the useful notion of essential self-adjointness.

**Definition 3.3** A symmetric operator $T$ is called essentially self- #-adjoint if its #-closure $\overline{\#T}$ is self- #-adjoint. If $T$ is #-closed, a subset $D \subset D(T)$ is called a core for $T$ if

$$ \overline{\#T} \uparrow \overline{D} = T. $$

If $T$ is essentially self-#-adjoint, then it has one and only one self-#-adjoint extension. The importance of essential self-#-adjointness is that one is often given a nonclosed symmetric operator $T$. If $T$ can be shown to be essentially self-#-adjoint, then there is uniquely associated to $T$ a self-adjoint operator $T = T^{**}$. Another way of saying this is that if $A$ is a self-#-adjoint operator, then to specify $A$ uniquely one need not give the exact domain of $A$ (which is often difficult), but just some #-core for $A$.

## Chapter V. Semigroups of operators on a non-Archimedean Banach spaces.
**Theorem 5.2.3** (spectral theorem-multiplication operator form) Let A be a bounded in $\mathbb{R}_c^\#$ self-#-adjoint operator on $\mathbf{H}^\#$, a $\#$-separable Hilbert space.

Then, there exist #-measures $\{\mu_n^\#\}_{n=1}^N$ with $N \in \mathbb{N}$ or $\{\mu_n^\#\}_{n=1}^{\infty}$ on $\sigma(A)$ and a

unitary operator $U: \mathbf{H}^\# \to \bigoplus_{n=1}^N L_2^\#(*\mathbb{R}_c^\#, d^\#\mu_n^\#)$ or $U: \mathbf{H}^\# \to \bigoplus_{n=1}^{\infty} L_2^\#(*\mathbb{R}_c^\#, d^\#\mu_n^\#)$

so that $(UAU^{-1}\psi)_n(\lambda) = \lambda\psi_n(\lambda)$

where we write an element $\psi \in \bigoplus_{n=1}^N L_2^\#(*\mathbb{R}_c^\#, d^\#\mu_n^\#)$ as an $N$-tuple $\langle\psi_1(\lambda), \dots, \psi_N(\lambda)\rangle$

or $\#$-tuple

This realization of A is called a spectral representation.

**Proof.** Use Lemma 5.2.2 to find the decomposition and then use Lemma 5.2.1 on each component.

This theorem tells us that every bounded self-#-adjoint operator is a multiplication operator on a suitable #-measure space; what changes as the operator changes are the underlying #-measures. Explicitly:

**Corolarly 5.2.1.** Let A be a bounded in $\mathbb{R}_c^\#$ self-adjoint operator on a $\#$-separable Hilbert space $\mathbf{H}^\#$. Then there exists a finite in $\mathbb{R}_c^\#$ measure space $\langle M, \mu^\# \rangle$, a bounded in $\mathbb{R}_c^\#$ function $F$ on $M$, and a unitary map, $U: \mathbf{H}^\# \to L_2^\#(M, d^\#\mu^\#)$ so that $(UAU^{-1}f)(m) = F(m)f(m)$.

**Proof** Choose the cyclic vectors $\phi_n$ so that $\|\phi_n\|_\# = 2^{-n}$. Let $M = \bigcup_{n=1}^{N^*} \mathbb{R}_c^\#$

i.e. the union of $N \in \mathbb{N}$ copies of $\mathbb{R}_c^\#$. Define $\mu$ by requiring that its restriction to the $n$-th copy of $\mathbb{R}_c^\#$ be $\mu_n$. Since $\mu(M) = \text{Ext}-\sum_{n=1}^N \mu_n^\#(*\mathbb{R}_c^\#) < \infty$, $\mu_n$ is finite in $\mathbb{R}_c^\#$. We also notice that this last theorem is essentially a rigorous form of the formaal Dirac notation. If we write $\phi_n = \phi(x; n)$, we see that in the “new representation defined by U” one has

$$ \langle \psi, \phi \rangle_{\#} = \text{Ext}-\sum_n \text{Ext}-\int d^\# \mu_n^\# \overline{\psi(\lambda; n)} \phi(\lambda; n) $$

and

$$ \langle \psi, A\phi \rangle_{\#} = \text{Ext}-\sum_n \text{Ext}-\int d^\# \mu_n^\# \overline{\psi(\lambda; n)} \lambda \phi(\lambda; n). $$

These are the Dirac type formulas familiar to physicists except that the formal sums of Dirac are replaced with integrals over spectral measures, where we define:

**Definition 5.2.3.** The #-measures $d^\#\mu_n$ are called spectral measures; they are just $d^\#\mu_\psi$ for suitable $\psi$.

**Remark 5.2.2.** Notice these #-measures are not uniquely determined.

We now investigate the connection between spectral measures and the spectrum.

**Definition 5.2.3.** If $\{\mu_n^\#\}_{n=1}^N, N \in \mathbb{N}$ is a family of #-measures, the support of $\{\mu_n^\#\}_{n=1}^N$ is the complement of the largest #-open set B with $\mu_n^\#(B) = 0$ for all $n \in \mathbb{N}$ so

$$ \text{supp}(\{\mu_n^\#\}_{n=1}^N) = \#-\overline{\bigcup_{n=1}^N \text{supp}(\mu_n^\#)}. \qquad (5.2.1) $$

**Proposition 5.2.1.** Let A be a self-#-adjoint operator and $\{\mu_n^\#\}_{n=1}^N, N \in \mathbb{N}$ a family of spectral #-measures. Then

$$ \sigma(A) = \text{supp}(\{\mu_n^\#\}_{n=1}^N). $$

**Definition 5.2.4.** Let $F$ be a $\mathbb{R}_c^\#$-valued function on a #-measure space $\langle M, \mu^\# \rangle$.

We say $\lambda$ is in the **essential range** of $F$ if and only if

$$ \mu^\#\{m | \lambda - \varepsilon < F(m) < \lambda + \varepsilon\} > 0. $$
§1.Semigroups on non-Archimedean Banach spaces and their generators.

A family of #-bounded operators {$T(t)|0 < t < \infty$} on external hyper infinite dimensional

non-Archimedean Banach space X endowed with $\mathbb{R}_{c,+}^{\#}$ - valued norm $\|\cdot\|_{\#}$ is called a strongly #-continuous semigroup if:

(a) $T(0) = I$

(b) $T(s)T(t) = T(s+t)$ for all $s, t \in \mathbb{R}_{c,+}^{\#}$

(c) For each $\varphi \in X, t \mapsto T(t)$ is #-continuous mapping.

We will see that strongly continuous semigroups are the “exponentials,”
$T(t) = \text{Ext}\text{-exp}(-tA)$, of a certain class of operators. .
We begin by studying a special class of semigroups:

**Definition 1.1.** A family {$T(t)|0 < t < \infty$} of bounded or hyper bounded operators on external hyper infinite dimensional Banach space $X$ is called a contraction semigroup if it is a strongly #-continuous semigroup and moreover $\|T(t)\|_{\#} < 1$ for all $t \in [0, \infty)$. Note that the all theorems about general strongly #-continuous semigroups are easy generalizations of the corresponding theorems for #-contraction semigroups. Thus, we study the special case first. We then briefly discuss the general theory and conclude the section by studying another special class, #-holomorphic semigroups.

**Proposition 1.1.** Let $T(t)$ be a strongly #-continuous semigroup on a non-Archimedean Banach space $X$ and set $A\varphi = \#\text{lim}_{r\to\# 0} A_r\varphi$ where $D(A) = \{\varphi | \#\text{lim}_{r\to\# 0} A_r\varphi \text{ exists}\}$. Then $A$ is

\#-closed and \#-densely defined. $A$ is called the infinitesimal generator of $T(t)$. We will also say that $A$ generates $T(t)$ and write $T(t) = \text{Ext}\text{-exp}(-tA)$.

**Proof.** Let $T(t)$ be a contraction semigroup on a Banach space $X$. We obtain the generator of $T(t)$ by \#-differentiation. Set $A_t = t^{-1}(I - T(t))$ and define

$$D(A) = \{\varphi | \#\text{lim}_{t \to \# 0} A_t \varphi \text{ exists}\}.$$

For $\varphi \in D(A)$, we define $A\varphi = \#\text{lim}_{t \to \# 0} A_t\varphi$. Our first goal is to show that $D(A)$ is \#-dense. For $\varphi \in X$, we set

$$\varphi_s = \text{Ext-} \int_0^s T(t) \varphi d^\#t. \qquad (2.1)$$

For any $r > 0$, we get

$$T(r)\varphi_s = \text{Ext-} \int_0^r T(t+r)\varphi d^\#t \qquad (2.2)$$

thus

$$ \begin{aligned} A_r \varphi_s = -\frac{1}{r} \left( \text{Ext-} \int_0^s [T(t+r)\varphi - T(t)\varphi] d^\# t \right) = & \\ -\frac{1}{r} \left( \text{Ext-} \int_s^{r+s} T(t) \varphi d^\# t \right) + \frac{1}{r} \left( \text{Ext-} \int_s^r T(t) \varphi d^\# t \right). & \end{aligned} \qquad (2.3) $$
for all $\varepsilon \approx 0, \varepsilon > 0$.

**Proposition 5.2.2.** Let $F$ be a bounded in $*\mathbb{R}_c^\# *\mathbb{R}_c^\#$-valued function on a #-measure space $\langle M, \mu^\# \rangle$. Let $T_f$ be the operator on $L_2^\#(M, d^\#\mu^\#)$ given by $(T_F g)(m) = F(m)g(m)$
Then $\sigma(T_F)$ is the essential range of $F$.

**Definition 5.2.5.** Let $A$ be a bounded in $*\mathbb{R}_c^\#$ self-#-adjoint operator on $H^\#$
Let $H_{pp}^\# = \{\psi|\mu_\psi^\# \text{ is pure point}\}, H_{ac}^\# = \{\psi|\mu_\psi^\# \text{ is absolutely #-continuous}\}$,
$H_{sing}^\# = \{\psi|\mu_\psi^\# \text{ is #-continuous singular}\}$.

We have thus proven.

**Theorem 5.2.4.** $H^\# = H_{pp}^\# \oplus H_{ac}^\# \oplus H_{sing}^\#$. Each of these subspaces is invariant under $A$.
$A \uparrow H_{pp}^\#$ has a #-complete set of eigenvectors, $A \uparrow H_{ac}^\#$ has only absolutely #-continuous
spectral #-measures and $A \uparrow H_{sing}^\#$ has only #-continuous singular spectral #-measures.

**Definition 5.2.6.** $\sigma_{pp}(A) = \{\lambda|\lambda \text{ is an eigenvalue of } A\}$,
$\sigma_{\#\text{cont}}(A) = \sigma(A \uparrow H_{\#\text{cont}}^\# = H_{\text{sing}}^\# \oplus H_{\text{ac}}^\#)$,
$\sigma_{\text{ac}}(A) = \sigma(A \uparrow H_{\text{ac}}^\#)$,
$\sigma_{\text{sing}}(A) = \sigma(A \uparrow H_{\text{sing}}^\#)$.

These sets are called the **pure point**, **#-continuous**, **absolutely #-continuous**, and **singular** (or **#-continuous singular**) **spectrum** respectively.

**Remark 4.2.2.** While it may happen that $\sigma_{ac}(A) \cup \sigma_{sing}(A) \cup \sigma_{pp}(A) \neq \sigma(A)$ this is only true because we did not define $\sigma_{pp}(A)$ as $\sigma(A \uparrow H_{pp}^\#)$ but rather as the actual set of eigenvalues.

**Proposition 4.2.3.** $\sigma_{\#\text{cont}}(A) = \sigma_{\text{ac}}(A) \cup \sigma_{\text{sing}}(A)$,
$\sigma(A) = \#\overline{\sigma_{\text{pp}}(A)} \cup \sigma_{\#\text{cont}}(A)$.

The sets need not be disjoint, however. The reader should be warned that $\sigma_{\text{sing}}(A)$
may have nonzero #-Lebesgue measure. For many purposes, breaking up the
spectrum in this way gives useful information.

Finally, we turn to the question of making canonical choices for the spectral
#-measures, a subject which goes under the title of “multiplicity theory.” We will
describe the basic results without proof:

§ 5.2.1. Multiplicity free operators

We must first ask when $A$ is unitariiy equivalent to multiplication by $x$ on $L_2^*( *\mathbb{R}_c^\#, d^\#\mu^\# )$,
that is, when only one spectral #-measure is needed. An symple examples tells us this
happens in the finite-dimensional case only when $A$ has no repeated eigenvalues, so
we define:

**Definition 5.2.7.** A bounded in $*\mathbb{R}_c^\#$ self-#-adjoint operator $A$ is called **multiplicity free** if and only if $A$ is unitarily equivalent to multiplication by $A$ on $L_2^*( *\mathbb{R}_c^\#, d^\#\mu^\# )$ for some #-measure $\mu^\#$.

One is interested in intrinsic characterizations of “multiplicity free” and there are
several:

**Theorem 5.2.5.** The following are equivalent:

(a) $A$ is multiplicity free.

(b) $A$ has a cyclic vector.

(c) $\{B|AB = BA\}$ is an abelian algebra.

#-Measure classes

Next we must ask about the nonuniqueness of the #-measure in the multiplicity free
From Eq.(2.3) one obtains $\#\lim_{r \to \#} A_r \varphi_s = -T(s)\varphi + \varphi$. Therefore, for each $\varphi \in X$

and $s > 0$, $\varphi_s \in D(A)$. Since $s^{-1}\varphi_s \to \# \varphi$ as $\to \# 0$, A is $\#$-densely defined.

Furthermore, if $\varphi \in D(A)$, then $A_r T(t)\varphi = T(t)A_r \varphi$, so $T(t) : D(A) \to D(A)$ and

$$ \frac{d^{\#}}{d^{\#}t} T(t)\varphi = -AT(t)\varphi = -T(t)A\varphi \quad (2.4) $$

A is also $\#$-closed, for if $\varphi_n \in D(A)$, $\#\lim_{n \to \infty\#} \varphi_n = \varphi$, and $\#\lim_{n \to \infty\#} A\varphi_n = \psi$, then

$$ \begin{aligned} \#\lim_{r \to \#} A_r \varphi &= \#\lim_{r \to \#} \#\lim_{n \to \infty\#} \left[ -\frac{1}{r} (T(r)\varphi_n - \varphi_n) \right] = \\ & \#\lim_{r \to \#} \#\lim_{n \to \infty\#} \frac{1}{r} \left( \text{Ext-} \int_s^r T(t) A \varphi_n d^\#t \right) = \\ & \#\lim_{r \to \#} \frac{1}{r} \left( \text{Ext-} \int_s^r T(t) \psi d^\#t \right) \end{aligned} \quad (2.5) $$

so $\varphi \in D(A)$ and $A\varphi = \psi$.

The formal Laplace transform

$$ \frac{1}{\lambda + A} = - \left( \text{Ext-} \int_{0}^{\infty} (\text{Ext-exp}(-\lambda t))(\text{Ext-exp}(-tA))d^\#t \right) \quad (2.6) $$

suggests that all $\mu \in \mathbb{C}_c^\#$ with $\text{Re}\,\mu < 0$ are in $\rho(A)$. This is in fact true and the formula (2.6) holds in the strong sense. For suppose that $\text{Re}\,\lambda > 0$. Then, since $\|\text{Ext-exp}(-tA)\| < 1$, the formula (2.7)

$$ R\varphi = \text{Ext-} \int_{0}^{\infty} (\text{Ext-exp}(-\lambda t))(\text{Ext-exp}(-tA)\varphi)d^\#t \quad (2.7) $$

defines a hyper bounded linear operator of $\#$-norm less than or equal to $(\text{Re}\,\lambda)^{-1}$. Moreover, for $r > 0$,

$$ \begin{aligned} A_r R\varphi = & -\frac{1}{r} \left( \text{Ext-} \int_{0}^{\infty} (\text{Ext-exp}(-\lambda t))(\text{Ext-exp}(-(t+r)A) - \text{Ext-exp}(-tA))\varphi d^\#t \right) = \\ & \frac{1 - \text{Ext-exp}(\lambda r)}{r} \left( \text{Ext-} \int_{0}^{\infty} (\text{Ext-exp}(-\lambda t))(\text{Ext-exp}(-tA))\varphi d^\#t \right) + \\ & \frac{\text{Ext-exp}(\lambda r)}{r} \left( \text{Ext-} \int_{0}^{r} (\text{Ext-exp}(-\lambda t))(\text{Ext-exp}(-tA))\varphi d^\#t \right) \end{aligned} \quad (2.8) $$

so as $r \to \# 0$, $A_r R\varphi \to \# (\varphi - \lambda R\varphi)$. Thus $R\varphi \in D(A)$ and $AR\varphi = \varphi - \lambda R\varphi$ which implies $(\lambda + A)R\varphi = \varphi$. In addition, for $\varphi \in D(A)$ we have $AR\varphi = RA\varphi$ since
case. Suppose $d^#\mu^#$ on $*\mathbb{R}_c^#$ is given and let $F$ be a #-measurable function which is positive and nonzero #-a.e. with respect to $\mu^#$ and locally $L_1^#(*\mathbb{R}_c^#, d^#\mu^#)$, that is,
$$ \int_{\Sigma} |F| d^#\mu^# < *\infty $$
for every compact set $\Sigma \subset *\mathbb{R}_c^#$. Then $d^#v = Fd^#\mu^#$ is a #-Borel #-measure and the map, $U : L_1^#(*\mathbb{R}_c^#, d^#v) \to L_1^#(*\mathbb{R}_c^#, d^#\mu^#)$ given by $(Uf)(\lambda) = U(\lambda f)$ is unitary (onto since $F \neq 0$ #-a.e.) and $\lambda(Uf) = U(\lambda f)$, Thus, an operator $A$ with a spectral representation in terms of $\pi$ could just as well be represented in terms of $v$. By the generalized Radon-Nikodym theorem, $d^#v = Fd^#\mu^#$ with $F$ #-a.e. nonzero if and only if $v^#$ and $\mu^#$ have the same sets of #-measure zero. This suggests the definition:

**Definition 4.2.8.** Two #-Borel #-measures $\mu^#$ and $v^#$ are called equivalent if and only if they have the same sets of #-measure zero. An equivalence class $\langle \mu^# \rangle$ is called a **#-measure class**.

Then, the nonuniqueness question is answered by:

**Proposition 4.2.7.** Let $\mu^#$ and $v^#$ be #-Borel #-measures on $*\mathbb{R}_c^#$ with bounded in $*\mathbb{R}_c^#$ support. Let $A_{\mu^#}$ be the operator on $L_2^#(*\mathbb{R}_c^#, d^#\mu^#)$ given by $(A_{\mu^#}f)(\lambda) = \lambda f(\lambda)$ and similarly for $A_{v^#}$ on $L_2^#(*\mathbb{R}_c^#, d^#v^#)$. Then $A_{\mu^#}$ and $A_{v^#}$ are unitarily equivalent if and only if $\mu^#$ and $v^#$ are equivalent #-measures.

## § 5.2.2. Operators of uniform multiplicity

If one wants a canonical listing of the eigenvalues of a matrix, it is natural to list all eigenvalues of multiplicity one, all eigenvalues of multiplicity two, etc. We thus need a way of saying that $A$ is an operator of uniform multiplicity two, three, etc. It is natural to take:

**Definition 5.2.9.** A bounded self-adjoint operator $A$ is said to be of uniform multiplicity $m \in *\mathbb{N}$ if $A$ is unitarily equivalent to multiplication by $\lambda$ on $Ext-\otimes L_2^#(*\mathbb{R}_c^#, d^#\mu^#)$ where there are $m$ terms in the external sum and $\mu^#$ is a fixed #-Borel #-measure.
That this is a good definition is shown by

**Proposition 5.2.8.** If $A$ is unitarily equivalent to multiplication by $\lambda$ on $Ext-\otimes L_2^#(*\mathbb{R}_c^#, d^#\mu^#)$ ($m$ times) and on $Ext-\otimes L_2^#(*\mathbb{R}_c^#, d^#v)$ ($n$ times), then $m-n$ and $\mu^#$ and $v^#$ are equivalent #-measures.

## § 5.2.3. Disjoint #-measure classes. The multiplicity theorem

In listing eigenvalues of multiplicity one, two, three, etc. in the finitedimensional case, we must add a requirement that prevents us from counting an eigenvalue of multiplicity

three once as an eigenvalue of multiplicity one and once as an eigenvalue of multiplicity

two. In the hyperfinite-dimensional case, we avoid this “error” by requiring the lists to be

disjoint. The analogous notion for #-measures is:

**Definition 5.2.10.** Two #-measure classes $\langle \mu^# \rangle$ and $\langle v^# \rangle$ are called disjoint if any $\mu_1^# \in \langle \mu^# \rangle$ and $v_1^# \in \langle v^# \rangle$ are mutually singular.

We can now state the basic theorem:
$$
\begin{align}
A\left(\operatorname{Ext}-\int_{0}^{\infty} (\operatorname{Ext}\operatorname{exp}(-\lambda t))(\operatorname{Ext}\operatorname{exp}(-tA))\varphi d^{\#}t\right) &= \nonumber \\
&\quad \operatorname{Ext}-\int_{0}^{\infty} (\operatorname{Ext}\operatorname{exp}(-\lambda t))A(\operatorname{Ext}\operatorname{exp}(-tA))\varphi d^{\#}t = \tag{2.9} \\
&\quad \operatorname{Ext}-\int_{0}^{\infty} (\operatorname{Ext}\operatorname{exp}(-\lambda t))(\operatorname{Ext}\operatorname{exp}(-tA))A\varphi d^{\#}t. \nonumber
\end{align}
$$

The first equality follows by approximation with external hyperfinite Riemann
sums (see [1]) from the facts that $(\operatorname{Ext}\operatorname{exp}(-\lambda t))(\operatorname{Ext}\operatorname{exp}(-tA))\varphi$ and
$A(\operatorname{Ext}\operatorname{exp}(-\lambda t))(\operatorname{Ext}\operatorname{exp}(-tA))$ are \#-integrable, $A$ is \#-closed. Thus, for $\varphi \in D(A)$,
$R(\lambda + A)\varphi = \varphi = (\lambda + A)R\varphi$ which implies that

$$
R = (\lambda + A)^{-1}. \tag{2.10}
$$

The properties of A which we have derived are also sufficient to guarantee that A generates a contraction semigroup. In fact, we only need information about real positive A.

**Theorem 1.1.** (Generalized Hille-Yosida theorem) A necessary and sufficient condition that a \#-closed linear operator $A$ on a Banach space $X$ generate a contraction semigroup is that

$$
(i) \quad (-\infty^{\#}, 0) \subset \rho(A)
$$

$$
(ii) \quad \|(\lambda + A)^{-1}\|_{\#} \text{ for all } \lambda > 0.
$$

Furthermore, if $A$ satisfies (i) and (ii), then the entire \#-open left half-plane is
contained in $\rho(A)$ and

$$
(\lambda + A)^{-1} \varphi = - \operatorname{Ext} - \int_{0}^{\infty} (\operatorname{Ext} \operatorname{exp}(-\lambda t))(\operatorname{Ext} \operatorname{exp}(-tA)) d^{\#}t \quad (2.11)
$$

for all $\varphi \in X$ and $\lambda$ with $\operatorname{Re}\lambda > 0$. Finally, if $T_1(t)$ and $T_2(t)$ are contraction semigroups generated by $A_1$ and $A_2$ respectively, then $T_2(t) \neq T_1(t)$ for some $t$ implies that
$A_1 \neq A_2$.

**Proof.** Since we showed above that conditions (i) and (ii) are necessary and that
(2.11)

holds, we need only show sufficiency. So, suppose that $A$ is a \#-closed operator on $X$
satisfying (i) and (ii). For $\lambda > 0$, define $A^{(\lambda)} = \lambda - \lambda^2(\lambda + A)^{-1}$. We will show that as
$\lambda \to \infty^{\#}$, $A^{(\lambda)} \to_{\#} A$ strongly on $D(A)$ and then construct $\operatorname{Ext}\operatorname{exp}(-tA)$ as the strong
\#-limit of the semigroups $\operatorname{Ext}\operatorname{exp}(-tA^{(\lambda)})$. For $\varphi \in D(A)$, $A^{(\lambda)}\varphi = \lambda(\lambda + A)^{-1}A\varphi$.
Moreover, by (ii),

$$
\#\text{-lim}_{\lambda \to \infty} [\lambda(\lambda + A)^{-1}\varphi - \varphi] = \#\text{-lim}_{\lambda \to \infty} [-(\lambda + A)^{-1}A\varphi] = 0. \quad (2.12)
$$

By condition (ii) the family ${\lambda(\lambda + A)^{-1}|\lambda > 0}$ is \#-uniformly hyperfinnitely bounded
in \#-norm, so since $D(A)$ is \#-dense, $\#\text{-lim}_{\lambda \to \infty} [\lambda(\lambda + A)^{-1}\psi] = \psi$ for all $\psi \in X$.
Thus $\#\text{-lim}_{\lambda \to \infty} A^{(\lambda)}\varphi = A\varphi$ for all $\varphi \in D(A)$. Since $A$ is hyperfinnitely bounded, the
semigroups $\text{Ext}\operatorname{exp}(-tA^{(\lambda)})$ can be defined by hyper infinite power series. Since
**Theorem 5.2.6.** (commutative multiplicity theorem) Let $A$ be abounded in $\mathbb{R}_c^\#\text{-#-adjoint operator on a Hilbert space } H^\#$. Then there is a decomposition $\mathrm{Ext}-\bigoplus_{m=1}^\infty H_m^\#$ so that

(a) $A$ leaves each $H_m^\#$ invariant.

(b) $A \upharpoonright H_m^\#$ has uniform multiplicity $m \in \mathbb{N}$.

(c) The #-measure classes $\langle\mu_m^\#\rangle$ associated with the spectral representation of $A \upharpoonright H_m^\#$ are mutually disjoint.

**Remark 5.2.3.** Moreover, the subspaces $\{H_m^\#\}_{m=1}^\infty$ (some of which may be zero) and the #-measure classes $\{\langle\mu_m^\#\rangle\}_{m=1}^\infty$ are uniquely determined by (a)-(c).

The spectral theorem with the multiplicity theory just described is thus one of those gems of mathematics: a structure theorem, that is, a theorem that describes all objects

of a certain sort up to a natural equivalence. Each bounded in $\mathbb{R}_c^\#$ self-#-adjoint operator $A$ is described by a family of mutually disjoint #-measure classes on $[-\|A\|_\#^*, \|A\|_\#]$; two operators are unitarily equivalent if and only if their spectral multiplicity #-measure classes are identical.

## § 5.3. Spectral projections.

In the last section, we constructed a functional calculus, $f \mapsto f(A)$ for any #-Borel function and any bounded in $\mathbb{R}_c^\#$ self-#-adjoint operator $A$. The most important functions gained in passing from the continuous functional calculus to the #-Borel functional calculus are the characteristic functions of sets.

**Definition 5.3.1.** Let $A$ be a bounded self-#-adjoint operator and $\Omega$ a #-Borel set of $\mathbb{R}_c^\#$. $P_\Omega = \chi_\Omega(A)$ is called a spectral projection of $A$.

As the definition suggests, $P_\Omega$ is an orthogonal projection since $\chi_\Omega = \chi_\Omega^2 = 1$ pointwise. The properties of the family of projections $\{P_\Omega|\Omega \text{ an arbitrary #-Borel set}\}$ is given by the following elementary translation of the functional calculus.

**Proposition 5.3.1.** The family $\{P_\Omega\}$ of spectral projections of a bounded self-#-adjoint operator $A$, has the following properties:

(a) Each $P_\Omega$ is an orthogonal projection.

(b) $P_\emptyset = 0$; $P_{(-a,a)} = I$ for some $a \in \mathbb{R}_c^\#$.

(c) If $\Omega = \mathrm{Ext}-\bigcup_{n=1}^\infty \Omega_n$ with $\Omega_n \cap \Omega_m = \emptyset$ for all $n \neq m$ then

$$P_\Omega = s\text{-#-}\lim_{N \to \infty} (\mathrm{Ext}-\sum_{n=1}^N P_{\Omega_n}). \quad (5.3.1)$$

(d) $P_{\Omega_1}P_{\Omega_2} = P_{\Omega_1 \cap \Omega_2}$.

**Definition 5.3.2.** A family of projections obeying (a)-(c) is called a projection-valued #-measure (p.v.-#-m.).

We remark that (d) follows from (a) and (c) by abstract considerations.

As one might guess, one can integrate with respect to a p.v.-#-m. If $P_\Omega$ is a p.v.-#-m., then $\langle\phi, P_\Omega\phi\rangle_\#$ is an ordinary #-measure for any $\phi$. We will use the symbol $d^\#\langle\phi, P_\lambda\phi\rangle_\#$ to mean integration with respect to this #-measure. By generalized Riesz lemma methods, there is a unique operator $B$ with $\langle\phi, B\phi\rangle_\# = \mathrm{Ext}-\int f(\lambda)d^\#\langle\phi, P_\lambda\phi\rangle_\#$.

**Theorem 5.3.1.** If $P_\Omega$ is a p.v.-#-m. and $f$ a bounded in $\mathbb{R}_c^\#$ #-Borel function on $\mathrm{supp}(P_\Omega)$, then there is a unique operator $B$ which we denote $\mathrm{Ext}-\int f(\lambda)d^\#P_\lambda$ so that

$$\langle\phi, B\phi\rangle_\# = \mathrm{Ext}-\int f(\lambda)d^\#\langle\phi, P_\lambda\phi\rangle_\# \quad (5.3.2)$$
$$
\begin{align}
\| \mathrm{Ext} \exp(-tA^{(\lambda)}) \|_{\#} &= \| (\mathrm{Ext} \exp(-\lambda t))(\mathrm{Ext} \exp(t\lambda^2(\lambda + A)^{-1})) \|_{\#} \le \nonumber \\
&\le (\mathrm{Ext} \exp(-\lambda t)) \left( \mathrm{Ext} \sum_{n=0}^{\infty} \frac{t^n \lambda^{2n}}{n!} \| (\lambda + A)^{-1} \|_{\#}^{n} \right) \le 1 \tag{2.13}
\end{align}
$$

they are contraction semigroups. For all $\mu, \lambda, t > 0$, and all $\varphi \in D(A)$, we have

$$
\begin{equation}
\begin{split}
& [\mathrm{Ext}\text{-exp}(-tA^{(\lambda)})]\varphi - [\mathrm{Ext}\text{-exp}(-tA^{(\mu)})]\varphi = \\
& \mathrm{Ext}\text{-}\int_0^t \frac{d^{\#}}{d^{\#}s} (\mathrm{Ext}\text{-exp}(-sA^{(\lambda)}))((\mathrm{Ext}\text{-exp}(-(t-s)A^{(\lambda)}))\varphi)d^{\#}s
\end{split}
\tag{2.14}
\end{equation}
$$

so,

$$
\begin{align*}
& \|[\mathrm{Ext}\text{-exp}(-tA^{(\lambda)})]\varphi - [\mathrm{Ext}\text{-exp}(-tA^{(\mu)})]\varphi\|_{\#} \le \\
& \mathrm{Ext}\text{-}\int_0^t \|(\mathrm{Ext}\text{-exp}(-sA^{(\lambda)}))((\mathrm{Ext}\text{-exp}(-(t-s)A^{(\lambda)})))\|_{\#} \|A^{(\mu)}\varphi - A^{(\lambda)}\varphi\|_{\#} d^{\#}s \le \tag{2.15} \\
& \le t \|A^{(\mu)}\varphi - A^{(\lambda)}\varphi\|_{\#}.
\end{align*}
$$

We have used the fact that $\mathrm{Ext}\exp(-tA^{(\lambda)})$ and $[\mathrm{Ext}\exp(-(t-s)A^{(\mu)})]$ commute
since $\{A^{(\lambda)}|\lambda > 0\}$ is a commuting family. Since we have proven above that
\#-lim$_{\lambda\to\infty}$\# $A^{(\lambda)}\varphi = A\varphi$, $\{\mathrm{Ext}\exp(-tA^{(\lambda)})\}$ is Cauchy as $\lambda \to \infty^\#$ for each $t > 0$ and
$\varphi \in D(A)$. Since $D(A)$ is \#-dense and the $\mathrm{Ext}\exp(-tA^{(\lambda)})$ are uniformly hyperfinitely
bounded, the same statement holds for all $\varphi \in X$. Now, define

$$
T(t)\varphi = \#\text{-lim}_{\lambda\to\infty}\#[\text{Ext}\exp(-tA^{(\lambda)})\varphi]. \quad (2.16)
$$

$T(t)$ is a semigroup of contraction operators since these properties are preserved
under strong \#-limits. The above inequality shows that the \#-convergence in Eq.(2.16)
is uniform for $t$ restricted to a hyperfinite interval, so $T(t)$ is strongly \#-continuous since
$\mathrm{Ext}\exp(-tA^{(\lambda)})$ is. Thus, $T(t)$ is a contraction semigroup.It remains to show that the
infinitesimal generator of $T(t)$, call it $\tilde{A}$, is equal to $A$. For all $t$
and $\varphi \in D(A)$,

$$
[\mathrm{Ext}\text{-exp}(-tA^{(\lambda)})\varphi] - \varphi = - \left[ \mathrm{Ext}\text{-}\left[ \int_0^t \mathrm{Ext}\text{-exp}(-sA^{(\lambda)}) \right] A^{(\lambda)} \varphi d^{\#}s \right] \quad (2.17)
$$

so, since \#-lim$_{\lambda\to\infty}$\# $A^{(\lambda)}\varphi = A\varphi$, we have

$$
T(t)\varphi - \varphi = - \left[ \mathrm{Ext} - \int_{0}^{t} T(s)A\varphi d^{\#}s \right]. \quad (2.18)
$$

Thus, $\tilde{A}_t \varphi \xrightarrow{\#} A\varphi$ as $t \xrightarrow{\#} 0$. Therefore $D(\tilde{A}) \supset D(A)$ and $\tilde{A} \upharpoonright D(A) = A$. For $\lambda > 0$,
$(\lambda + A)^{-1}$ exists by hypothesis and $(\lambda + \tilde{A})^{-1}$ exists by the necessity part of the
theorem.

§2 Hypercontractive semigroups

In the previous section we discussed $\mathcal{L}_{\#}^p$-contractive semigroups. In this section we will
prove a self-adjointness theorem for operators of the form $A + V$ where $V$ is a
**Theorem 5.3.2.** (spectral theorem-p.v.#-m. form) There is a one-one correspondence between (bounded) self-#-adjoint operators A and (bounded) projection valued #-measures $\{P_\Omega\}$ given by

$$A \mapsto \{P_\Omega\} = \{\chi_\Omega(A)\} \qquad (5.3.3)$$

and

$$\{P_\Omega\} \mapsto A = \mathrm{Ext} - \int \lambda d^\# P_\lambda. \qquad (5.3.4)$$

Spectral projections can be used to investigate the spectrum of A.

**Proposition 5.3.1.** $\lambda \in \sigma(A)$ if and only if $P_{(\lambda-\varepsilon, \lambda+\varepsilon)}(A)$ for any $\varepsilon > 0$.

The essential element of the proof is that $\|(A - \lambda)^{-1}\|_{\#} = [\mathrm{dist}(\lambda, \sigma(A))]^{-1}$.

This suggests that we distinguish between two types of spectrum.

**Definition 5.3.3.** We say that (i) $\lambda \in \sigma_{ess}(A)$, the essential spectrum of A if and only if $P_{(\lambda-\varepsilon, \lambda+\varepsilon)}(A)$ is hyper infinite dimensional for all $\varepsilon > 0$.

(ii) If $\lambda \in \sigma(A)$ but $P_{(\lambda-\varepsilon, \lambda+\varepsilon)}(A)$ is hyperfinite dimensional for some $\varepsilon > 0$, we say $\lambda \in \sigma_{disc}(A)$, the discrete spectrum of A. P is hyper infinite dimensional means $\mathrm{Ran}(P)$ is hyper infinite dimensional.

Thus, we have a second decomposition of $\sigma(A)$. Unlike the first, it is a decomposition into two necessarily disjoint subsets. We note that $\sigma_{disc}$ is not necessarily #-closed, but notice that.

**Theorem 5.3.3** $\sigma_{ess}(A)$ is always #-closed.

**Proof** Let $\lambda_n \to \lambda$ with each $\lambda_n \in \sigma_{ess}(A)$. Since any #-open interval $I$ about $\lambda$ contains an interval about some $\lambda_n$, $P_I(A)$ is hyper infinite dimensional.

The following three theorems give alternative descriptions of $\sigma_{disc}$ and $\sigma_{ess}$;

**Theorem 5.3.4** $\lambda \in \sigma_{disc}$ if and only if both the following hold:

(a) $\lambda$ is an #-isolated point of $\sigma(A)$ that is, for some $\varepsilon \approx 0$, $(\lambda - \varepsilon, \lambda + \varepsilon) \cap \sigma(A) = \{\lambda\}$.

(b) $\lambda$ is an eigenvalue of hyperfinite multiplicity, i.e., $\{\psi|A\psi = \lambda\psi\}$ is hyperfinite dimensional.

**Theorem 5.3.5** $\lambda \in \sigma_{ess}$ if and only if one or more of the following holds:

(a) $\lambda \in \sigma_{\#\mathrm{cont}}(A) \leftrightarrow \sigma_{\mathrm{ac}}(A) \cup \sigma_{\mathrm{sing}}(A)$.

(b) $\lambda$ is a #-limit point of $\sigma_{\mathrm{pp}}(A)$.

(c) $\lambda$ is an eigenvalue of hyper infinite multiplicity.

**Theorem 5.3.6** (Generalized Weyl's criterion) Let A be a bounded in $\mathbb{R}_c^\#$ self-#-adjoint operator. Then (i) $\lambda \in \sigma(A)$ if and only if there exists $\{\psi_n\}_{n=1}^\infty$ with $\|\psi_n\|_\# = 1$ and $\#\lim_{n \to \infty} \|(A - \lambda)\psi_n\|_\# = 0$.

(ii) $\lambda \in \sigma_{ess}(A)$ if and only if the above $\{\psi_n\}$ can be chosen to be orthogonal.

As one might guess, the essential spectrum cannot be removed by essentially hyperfinite dimensional perturbations. In Section 5.4, we will prove a general theorem which implies that $\sigma_{ess}(A) = \sigma_{ess}(B)$ if $A \setminus B$ is #-compact.

Finally, we discuss one useful formula relating the resolvent and spectral projections. It is a matter of computation to see that

$$f_\varepsilon(x) \to \# \begin{cases} 0 & \text{if } x \notin [a, b] \\ 1/2 & \text{if } x = a \lor x = b \\ 1 & \text{if } x \in (a, b) \end{cases}$$
multiplication operator and A generates an $\mathcal{L}_{\#}^p$-contractive semigroup that satisfies a strong additional property.

**Definition 2.1.** Let $\langle M, \mu^# \rangle$ be a \#-measure space with $\mu^#(M) = 1$ and suppose that A is a positive self-adjoint operator on $\mathcal{L}_{\#}^2(M, d^#\mu^#)$. We say that $\mathrm{Ext}\text{-exp}(-tA)$ is a hypercontractive semigroup if:

(i) $\mathrm{Ext}\text{-exp}(-tA)$ is $\mathcal{L}_{\#}^p$-contractive;

(ii) for some $b > 2$ and some constant $C_b$, there is a $T > 0$ so that
$$ \|\mathrm{Ext}\text{-exp}(-tA)\varphi\|_b \le C_b \|\varphi\|_2 \quad \text{for all } \varphi \in \mathcal{L}_{\#}^2(M, d^#\mu^#). $$

By Theorem X.55, condition (i) implies that $\mathrm{Ext}\text{-exp}(-tA)$ is a strongly \#-continuous contraction semigroup for all $p < \infty^#$. Holder's inequality shows that

$$ \|\cdot\|_q \le \|\cdot\|_p \qquad (1) $$

if $p \ge q$. Thus the $\mathcal{L}_{\#}^p$-Spaces are a nested family of spaces which get smaller as $p$ gets larger; this suggests that (ii) is a very strong condition. The following proposition shows

that $b$ plays no special role.

**Proposition 2.1.** Let $\mathrm{Ext}\text{-exp}(-tA)$ be a hypercontractive semigroup on $\mathcal{L}_{\#}^2(M, d^#\mu^#)$. Then for all $p, q \in (1, \infty^#)$, there is a constant $C_{pq}$ and a $t_{pq} > 0$ so that if $t > t_{pq}$ then
$$ \|\mathrm{Ext}\text{-exp}(-tA)\varphi\|_p \le C_{pq} \|\varphi\|_q \quad \text{for all } \varphi \in \mathcal{L}_{\#}^q. $$

**Proof.** The case where $p < q$ follows immediately from (i) and (1). So suppose that $p > q$. Since $\mathrm{Ext}\text{-exp}(-tA) : \mathcal{L}_{\#}^2 \to \mathcal{L}_{\#}^b$ and $\mathrm{Ext}\text{-exp}(-tA) : \mathcal{L}_{\#}^{\infty^#} \to \mathcal{L}_{\#}^{\infty^#}$, the generalized Riesz-Thorin theorem implies that there is a constant $C$ so that for all $r \ge 2$, $\|\mathrm{Ext}\text{-exp}(-tA)\varphi\|_r \le C\|\varphi\|_{br/2}$. We now consider two cases. First, if $q \ge 2$ we choose $n$ large enough so that $2(b/2)^n > p$. Then $\|\mathrm{Ext}\text{-exp}(-nTA)\varphi\|_{2(b/2)^n} \le C\|\varphi\|_2$ so the conclusion follows if $2 < q, p > 2(b/2)^n$, by using (1), and hypothesis (i). If $1 < q < 2$, then we choose $n$ large enough so that $2(b/2)^n > p$ and $q > c$ where $c^{-1} + (2(b/2)^n)^{-1} = 1$. Since A is self-adjoint and $\mathrm{Ext}\text{-exp}(-nTA)\varphi$ is a bounded or hyper bounded map from $\mathcal{L}_{\#}^2$ to $\mathcal{L}_{\#}^{2(b/2)^n}$, $(\mathrm{Ext}\text{-exp}(-nTA))^* = \mathrm{Ext}\text{-exp}(-nTA)$ is a bounded or hyper bounded map from $\mathcal{L}_{\#}^c$ to $\mathcal{L}_{\#}^2$. Thus $\mathrm{Ext}\text{-exp}(-2nTA)$ is a bounded or hyper bounded map from $\mathcal{L}_{\#}^c$ to $\mathcal{L}_{\#}^{2(b/2)^n}$. Since $c < q < p < 2(b/2)^n$, (1) implies the proposition.

**Theorem 2.1.** The operator $-\frac{1}{2}d^{\#2}/d^#x^2 + xd^#/d^#x$ on $\mathcal{L}_{\#}^2(*\mathbb{R}_{\#}^c, \pi_{\#}^{-1/2}\mathrm{Ext}\text{-exp}(-x^2)d^#x)$ is positive and essentially self-adjoint on the set of hyperfinite linear combinations of Hermite polynomials, and generates a hypercontractive semigroup. As a preparation for our main theorem, we prove the following result.

**Theorem 2.2** Let $\langle M, \mu \rangle$ be a \#-measure space with $\mu(M) = 1$ and let $H_0$ be the generator of a hypercontractive semigroup on $\mathcal{L}_{\#}^2(M, d\mu)$. Let $V$ be a real-valued measurable function on $\langle M, \mu^# \rangle$ such that $V \in \mathcal{L}_{\#}^p(M, d^#\mu^#)$ for all $p \in [1, \infty^#)$ and $\mathrm{Ext}\text{-exp}(-tV) \in \mathcal{L}_{\#}^1(M, d^#\mu^#)$ for all $t > 0$. Then $H_0 + V$ is essentially self-\#-adjoint on $C^{\infty^#}(H_0) \cap D(V)$ and is bounded below. $C^{\infty^#}(H_0) = \cap_{p \in \mathbb{N}^#} D(H_0^p)$

# Chapter VI. Singular Perturbations of Selfadjoint Operators on a non-Archimedean Hilbert space.

## §1. Introduction
if $\varepsilon \to 0$, where

$$f_{\varepsilon}(x) = (2\pi\#i)^{-1} \left( \mathrm{Ext} - \int_a^b [(x - \lambda - i\varepsilon)^{-1} - (x - \lambda + i\varepsilon)^{-1}] d^{\#}\lambda \right). \quad (5.3.5)$$

Moreover, $|f_{\varepsilon}(x)|$ is bounded in $x \in \mathbb{R}_c^{\#}$ uniformly in $\varepsilon \approx 0$, so by the functional calculus, one obtains that.

**Theorem 5.3.7 (Generalized Stone's formula)** Let A be a bounded in $\mathbb{R}_c^{\#}$ self-#-adjoint operator. Then

$$ \begin{aligned} \mathrm{s-lim}_{\varepsilon \to 0} (2\pi\#i)^{-1} \left( \mathrm{Ext} - \int_a^b [(A - \lambda - i\varepsilon)^{-1} - (A - \lambda + i\varepsilon)^{-1}] d^{\#}\lambda \right) &= \tag{5.3.6} \\ &= \frac{1}{2} [P_{[a,b]} + P_{(a,b)}]. \end{aligned} $$

## § 5.4. The #-continuous functional calculus related to unbounded in $\mathbb{R}_c^{\#}$ self-#-adjoint operators

In this section we will show how the spectral theorem for bounded in $\mathbb{R}_c^{\#}$ self-#-adjoint operators which we developed in § 4.3 can be extended to unbounded in $\mathbb{R}_c^{\#}$ self-#-adjoint operators. To indicate what we are aiming for, we first prove the following:

**Proposition 5.4.1.** Let $\langle M, \mu^{\#} \rangle$ be a #-measure space with $\mu^{\#}$ a hyperfinite #-measure. Suppose that $f$ is a #-measurable, $\mathbb{R}_c^{\#}$-valued function on $M$ which is finite or hyperfinite a.e. $\mu^{\#}$. Then the operator $T_f: \varphi \to f\varphi$ on $L_2^{\#}(M, d^{\#}\mu^{\#})$ with domain

$$D(T_f) = \{\varphi | f\varphi \in L_2^{\#}(M, d^{\#}\mu^{\#})\} \quad (5.4.1)$$

is self-#-adjoint and $\sigma(T_f)$ is the essential range of $T_f$.

**Proof** $T_f$ is clearly symmetric. Suppose that $\psi \in D(T_f^*)$ and let

$$\chi_N = \begin{cases} 1 & \text{if } |f(m)| \le N \\ 0 & \text{otherwise} \end{cases}$$

Then, using the generalized monotone #-convergence theorem,

$$\| T_f^* \psi \|_\# = \#\text{-lim}_{N \to \infty} \| \chi_N T_f^* \psi \|_\# = \#\text{-lim}_{N \to \infty} \left( \sup_{\|\varphi\|_\# = 1} |\langle \varphi, \chi_N T_f^* \psi \rangle_\#| \right) =$$

$$\#\text{-lim}_{N \to \infty} \left( \sup_{\|\varphi\|_\# = 1} |\langle \chi_N T_f \varphi, \psi \rangle_\#| \right) = \#\text{-lim}_{N \to \infty} \left( \sup_{\|\varphi\|_\# = 1} |\langle \varphi, \chi_N f \psi \rangle_\#| \right) =$$

$$\#\text{-lim}_{N \to \infty} \| \chi_N f \psi \|_\#$$

Thus, $f\psi \in L_2^{\#}(M, d^{\#}\mu^{\#})$, so $\psi \in D(T_f)$ and therefore $T_f$ is self-#-adjoint. That $\sigma(T_f)$ is the essential range of $f$ follows as in the bounded case.

With more information about $f$, one can say something about the domains on which $T_f$ is essentially self-#-adjoint:

**Proposition 5.4.2.** Let $f$ and $T_f$ obey the conditions in Proposition 4.4.1. Suppose in addition that $f \in L_p^{\#}(M, d^{\#}\mu^{\#})$ for $2 < p < \infty$. Let $D$ be any #-dense set in $L_q^{\#}(M, d^{\#}\mu^{\#})$ where $q^{-1} + p^{-1} = 1/2$. Then $D$ is a #-core for $T_f$.

**Proof** Let us first show that $L_q^{\#}$ is a #-core for $T_f$. By the generalized Holder's inequality $\|g\|_{\#2} \le \|1\|_{\#p} \cdot \|g\|_{\#q}$, and $\|fg\|_{\#2} \le \|f\|_{\#p} \cdot \|g\|_{\#q}$ so $L_p^{\#} \subset D(T_f)$.
We study the sum $A + B$ of two #-selfadjoint operators on a non-Archimedean Banach spaces, and we find sufficient conditions for $C = A + B$ to be #-selfadjoint. Our technique is to approximate $B$ by a hyperinfinite sequence of bounded #-selfadjoint operators $B_n, n \in *\mathbb{N}$ and so to approximate $C$ by #-selfadjoint operators $C_n = A + B_n$. We answer three questions separately:

1. When do the operators $C_n$ have a #-lim $C$? 2. When is $C$ a #-selfadjoint operator?

3. When is $C = A + B$?

In Theorem 8 we give a set of estimates on the relative size of $A$ and $B$ which ensure a positive answer to all three questions. Hence these estimates show that $A + B = C$ is #-selfadjoint. In another paper [5], we use Theorem 2.8 to prove the existence of a self-interacting, causal quantum field in 4-dimensional space-time. Formally this field theory is Lorentz covariant and has non-trivial scattering; this application was the motivation for the present work.

In order to investigate the meaning of #-lim$_{n \to *\infty}$ $C_n$, we give a new definition for the strong #-convergence of a hyperinfinite sequence of operators. Consequences of this definition

are worked out in Section 2. In Section 3 we give estimates on operators $C_n$ which are sufficient to ensure that the #-lim$_{n \to *\infty}$ $C_n = C$ exists and that $C$ is maximal symmetric or #-selfadjoint. This result is given in Theorem 5 and Corollary 6.

In Section 4 we investigate whether #-lim$_{n \to *\infty}$ $C_n = C$ is equal to $A + B$.

We combine this work in Theorem 8, our second main theorem, where $B$ is a singular, but nearly positive #-selfadjoint perturbation of a positive #-selfadjoint operator $A$. To illustrate this theorem, let $A \ge I$ and let $B$ be essentially #-selfadjoint on

$$D^{\#} = \bigcap_{n \in *\mathbb{N}} D(A^n). \qquad (1.0)$$

Assume now that, for some $\beta > 0$ and some $\alpha$,

$$A^{-(1-\beta)}BA^{-(1-\beta)} \text{ and } A^{\beta}BA^{\alpha} \qquad (1.1)$$

are #-densely defined, bounded operators. Also, for some positive $a, \varepsilon \in *\mathbb{R}_{c^+}^\#$ satisfying $2a + \varepsilon < 1$, suppose that there is a constant $b \in *\mathbb{R}_c^\#$ such that, as bilinear forms on $D \times D$,

$$0 \le aA + B + b \qquad (1.2)$$

and

$$0 \le \varepsilon A^2 + [A^{1/2}, [A^{1/2}, B]] + b. \qquad (1.3)$$

Then $A + B$ is #-selfadjoint.

We see from this example that neither the operator $B$ nor the bilinear form $B$ need be bounded relative to $A$.

While it may not appear evident, the conditions (1.1)-(1.3) are closely related to a more easily understandable estimate on $D^\# \times D^\#$,

$$A^2 + B^2 c(A + B)^2 + c. \qquad (1.4)$$

In fact, estimates (1.1)-(1.3) are chosen because they allow us not only to prove (1.4), but also the similar inequality where $B$ is replaced by $B_n$.

Let us now see that if $A + B$ is #-selfadjoint, then (1.4) must hold for every vector in $D(A + B) = D(A) \cap D(B)$.
Moreover, if $g \in D(T_f)$ let $g_n, n \in *\mathbb{N}$ be that function which is zero where $|g(m)| > n$ and equal to $g$ otherwise. By the generalized dominated convergence theorem, $g_n \to_* g$ and $fg_n \to_* fg$ in $L_2^\#$. Since each $g_n$ is in $L_q^\#$, we conclude that $L_q^\#$ is a #-core for $T_f$. Now let $D$ be #-dense in $L_q^\#$ and let $g \in L_q^\#$. Find $g_n \in D$ with $g_n \to_* g$ in $L_q^\#$. Since $\|g_n - g\|_{\#2} \le \|1\|_{\#p} \cdot \|g_n - g\|_{\#q}$ and $\|T_f(g_n - g)\|_{\#2} \le \|f\|_{\#p} \cdot \|g_n - g\|_{\#q}$, $g \in *\overline{D(T_f \upharpoonright D)}$.

Thus $L_q^\# \subset D(T_f \upharpoonright D)$ so $D$ is a #-core. Unless $f \in L_{\infty}^\#(M, d^\#\mu^\#)$ the operator $T_f$ described in Propositions 5.4.1 and 5.4.2 will be unbounded.

Thus, we have found a large class of unbounded self-#-adjoint operators. In fact, we have found them all.

**Theorem 5.4.1.** (spectral theorem-multiplication operator form) Let $A$ be a self-#-adjoint operator on a $*\infty$-dimensional a non-Archimedean Hilbert space $\mathbf{H}^\#$ with domain $D(A)$. Then there is a #-measure space $\langle M, \mu^\# \rangle$ with $\mu^\#$ a hyperfinite #-measure, a unitary operator $U : \mathbf{H}^\# \to L_2^\#(M, d^\#\mu^\#)$, and a $*\mathbb{R}_c^\#$-valued function $f$ on $M$ which is finite or hyperfinite $\mu^\#$-a.e. so that

(a) $\psi \in D(A)$ if and only if $f(\cdot)(U\psi)(\cdot) \in L_2^\#(M, d^\#\mu^\#)$.

(b) If $\varphi \in U[D(A)]$, then $(UAU^{-1}\varphi)(m) = f(m)\varphi(m)$.

**Proof** It easily verify that $A + i$ and $A - i$ are one to one correspondence and $\mathrm{Ran}(A \pm i) = \mathbf{H}^\#$. Since $A \pm i$ are #-closed, $(A \pm i)^{-1}$ are #-closed and therefore bounded in $*\mathbb{R}_c^\#$. Note that the operators $(A + i)^{-1}$ and $(A - i)^{-1}$ commute. The equality $\langle (A - i)\psi, (A + i)^{-1}(A + i)\varphi \rangle_\# = \langle (A + i)^{-1}(A - i)\psi, (A + i)\varphi \rangle_\#$ and the fact that $\mathrm{Ran}(A \pm i) = \mathbf{H}^\#$ shows that $((A + i)^{-1})^* = (A - i)^{-1}$. Thus the operator $(A + i)^{-1}$ is normal.

We now use the easy extension of the spectral theorem for bounded in $*\mathbb{R}_c^\#$ self-#-adjoint operators to bounded in $*\mathbb{R}_c^\#$ normal operators. The proof of this extension is a straightforward. We conclude that there is a #-measure space $\langle M, \mu^\# \rangle$ with $\mu^\#$ a hyperfinite #-measure, a unitary operator $U : \mathbf{H}^\# \to L_2^\#(M, d^\#\mu^\#)$, and a #-measurable, bounded, in $*\tilde{\mathbb{R}}_c^\# *\tilde{\mathbb{C}}_c^\#$-valued function $g(m)$ so that $U(A + i)^{-1}U^{-1}\varphi(m) = g(m)\varphi(m)$ for all $\varphi \in L_2^\#(M, d^\#\mu^\#)$. Since $\mathrm{Ker}((A + i)^{-1})$ is empty, $g(m) \ne 0$ a.e. $\mu^\#$, so the function $f(m) = g^{-1}(m) - i$ is hyperfinite a.e. $\mu^\#$. Now, suppose $\psi \in D(A)$. Then $\psi = (A + i)^{-1}\varphi$ for some $\varphi \in \mathbf{H}^\#$ and $U\psi = gU\varphi$. Since $fg$ is bounded in $*\mathbb{R}_c^\#$, we conclude that $f(U\psi) \in L_2^\#(M, d^\#\mu^\#)$. Conversely, if $f(U\psi) \in L_2^\#(M, d^\#\mu^\#)$, then there is a $\varphi \in \mathbf{H}^\#$ so that $U\varphi = (f + i)U\psi$. Thus, $gU\varphi = g(f + i)U\psi = U\psi$, so $\psi = (A + i)^{-1}\varphi$ which shows that $\psi \in D(A)$. This proves (a).

To prove (b) notice that if $\psi \in D(A)$ then $\psi = (A + i)^{-1}\varphi$ for some $\varphi \in \mathbf{H}^\#$ and $A\psi = \varphi - i\psi$. Therefore, $(UA\psi)(m) = (U\varphi)(m) - i(U\psi)(m) = (g^{-1}(m) - i)(U\psi)(m) = f(m)(U\psi)(m)$. Finally, if $\mathrm{Im}(f) > 0$ on a set of nonzero Lebesgue #-measure, there is a bounded in $*\mathbb{R}_c^\#$ set $B$ in the upper half plane so that $S = \{x|f(x) \in B\}$ has nonzero Lebesgue #-measure. If $\chi(x)$ is the characteristic function of $S$ then $f\chi \in L_2^\#(M, d^\#\mu^\#)$ and $\mathrm{Im}\langle\chi, f\chi\rangle > 0$. This contradicts the fact that multiplication by $i$ is self-adjoint (since it is unitarily equivalent to $A$). Thus $f$ is $*\mathbb{R}_c^\#$-valued function.

There is a natural way to define functions of a self-#-adjoint operator by using the above theorem. Given a bounded in $*\mathbb{R}_c^\#$ #-Borel function $h$ on $*\mathbb{R}_c^\#$ we define

$$h(A) = UT_{h(f)}U^{-1} \qquad (5.4.2)$$
**Proposition 1.1.** Let A and B be #-closed operators. Then A + B is #-closed if and only if there is a constant c ∈ *R# c such that for all ψ ∈ D(A + B)

$$
\|A\psi\|_{\#} + \|B\psi\|_{\#} \le \|(A+B)\psi\|_{\#} + c\|\psi\|_{\#} \quad (1.5)
$$

and (1.5) is equivalent to (1.4) on D(A + B) × D(A + B).

**Proof:** Certainly (1.5) implies that A + B is #-closed. Conversely, assume that A + B is #-closed and introduce the #-norms on D(A + B) = D(A) ∩ D(B),

$$
\|\psi\|_{\#1} \triangleq \|\psi\|_{\#} + \|A\psi\|_{\#} + \|B\psi\|_{\#} \quad (1.6)
$$

and

$$
\|\psi\|_{\#2} \triangleq \|\psi\|_{\#} + \|(A+B)\psi\|_{\#} \quad (1.7)
$$

Then D(A + B), ||·||_{#2} is a non-Archimedean Banach space because A + B is #-closed.
The identity map from D(A + B), ||·||_{#2} to D(A + B), ||·||_{#1} has a #-closed graph because
A, B, and A + B are c#-losed. By the #-closed graph theorem, the identity map is
#-continuous; hence

$$
\|\psi\|_{\#1} \le c \|\psi\|_{\#2}. \qquad (1.7')
$$

**Proposition 1.2.** Let $A \ge I, B$ be #-selfadjoint operators with $D^\# \subset D(B)$ and suppose (1.2) and (1.3) hold. Then (1.4) is valid on $D^\# \times D^\#$.

**Proof** The operators $A^2, B^2, AB, BA$, and $A^{1/2}BA^{1/2}$ define bilinear forms on $D^\# \times D^\#$. Using (1.2) and (1.3), we have the inequality:

$$
A^2 + B^2 = (A + B)^2 - 2A^{1/2}BA^{1/2} - [A^{1/2}, [A^{1/2}, B]] \le (A + B)^2 + (2a + \epsilon)A^2 + 2Ab + b \\
\text{which establishes (1.4).}
$$

§2. Strong #-Convergence of Operators

Let $\mathcal{L}(C)$ be the graph of the operator $C$. For any hyper infinite sequence $\{C_n\}$, $n \in *\mathbb{N}$
of #-densely defined operators we define

$$
\mathcal{L}_{*\infty}(C) = \{\phi, \chi | \phi = \#\text{-lim}_{n \to *\infty} \phi_n, \phi_n \in D(C_n), \chi = \#\text{-lim}_{n \to *\infty} C_n \phi_n\}. \quad (8)
$$

In general, $\mathcal{L}_{*\infty}$ will not be the graph of an operator. If the hyper infinite sequence
$\{C_n^*\}$, $n \in *\mathbb{N}$#-converges strongly on a #-dense domain $D$ to an operator $C^*$, namely,

$$
C^* \psi = \#\text{-lim}_{n \to *\infty} C_n^* \psi, \psi \in D,
$$

then $\mathcal{L}_{*\infty}$ is the graph of some operator $C^*$. In particular, if each $C_n$ is self #-adjoint, and if the $C_n$ #-converge on a #-dense set $D$ to an operator $C$ defined on $D$, then $\mathcal{L}_{*\infty} = \mathcal{L}_{*\infty}(C_{*\infty})$ and $C_{*\infty}$ is a symmetric extension of $C$.

**Definition 2.1.** $G$ #-CONVERGENCE. The hyper infinite sequence of operators $C_n, n \in *\mathbb{N}$ #-converge strongly to $C_{*\infty}$ in the sense of graphs, written

$$
C_n \to_{\#G} C_{*\infty} \qquad (8')
$$

if $\mathcal{L}_{*\infty}$ is the graph of a #-densely defined operator $C_{*\infty}$.

**Remark 2.1.** Note that for a hyper infinite sequence of uniformly bounded operators
$\{C_n^*\}_{n \in *\mathbb{N}}$ such that $C_n \to_{\#G} C_{*\infty}$, $C_{*\infty}$ is the usual strong #-limit of the operators
$C_n, n \in *\mathbb{N}$ and is everywhere defined.

**Definition 2.2.R** #-CONVERGENCE. Let the resolvents $R_n(z) = (C_n - z)^{-1}, n \in *\mathbb{N}$
exist for some $z \in *\mathbb{C}^#$, and be uniformly bounded in $n$. The operators $C_n$ #-converge
strongly to $C_{*\infty}$ in the sense of resolvents, written
where $T_{h(f)}$ is the operator on $L_2^\#(M, d^\#\mu^\#)$ which acts by multiplication by the function $h(f(m))$. Using this definition the following theorem follows easily from Theorem 4.4.1.

**Theorem 5.4.2.** (spectral theorem-functional calculus form) Let $A$ be a self-#-adjoint operator on $\mathbf{H}^\#$. Then there is a unique map $\hat{\phi}$ from the bounded #-Borel functions on $\#\mathbb{R}_c^\#$ into $\mathcal{L}(\mathbf{H}^\#)$ so that

(a) $\hat{\phi}$ is an algebraic $\#$-homomorphism.

(b) $\hat{\phi}$ is $\#$-norm $\#$-continuous, that is, $\|\hat{\phi}(h)\|_{\mathcal{L}(\mathbf{H}^\#)} \le \|h\|_{\#\infty}$

(c) Let $h_n(x), n \in \#\mathbb{N}$ be a hyper infinite sequence of bounded in $\#\mathbb{R}_c^\#$ $\#$-Borel functions with $\#\lim_{n \to \#\infty} h_n(x) = x$

for each $x$ and $|h_n(x)| \le |x|$ for all $x$ and $n \in \#\mathbb{N}$. Then, for any $\psi \in D(A)$, $\#\lim_{n \to \#\infty} \hat{\phi}(h_n)\psi = A\psi$.

(d) If $h_n(x) \to_\# h(x)$ pointwise and if the hyper infinite sequence $\|h_n\|_{\#\infty}, n \in \#\mathbb{N}$ is bounded in $\#\mathbb{R}_c^\#$, then $\hat{\phi}(h_n) \to_\# \hat{\phi}(h)$ strongly.

In addition:

(e) If $A\psi = \lambda\psi$ then $\hat{\phi}(h) = h(\lambda)\psi$.

(f) If $h \ge 0$, then $\hat{\phi}(h) \ge 0$.

The functional calculus is very useful. For example, it allows us to define the exponential $\mathrm{Ext}\text{-}\exp(itA)$ and prove easily many of its properties as a function of $t$ (see the next section). In the case where $A$ is bounded in $\#\mathbb{R}_c^\#$ we do not need the functional calculus to define the exponential since we can define $\mathrm{Ext}\text{-}\exp(itA)$ by the power series which $\#$-converges in $\#$-norm.

The functional calculus is also used to construct spectral $\#$-measures and can be used to develop a multiplicity theory similar to that for bounded self-$\#$-adjoint operators.

A vector $\psi \in \mathbf{H}^\#$ is said to be cyclic for $A$ if $\{g(A)\psi | g \in C^{\#\infty}(\#\mathbb{R}_c^\#)\}$ is $\#$-dense in $\mathbf{H}^\#$.

If $\psi$ is a cyclic vector, then it is possible to represent $\mathbf{H}^\#$ as $L_2^\#(\#\mathbb{R}_c^\#, d^\#\mu_\psi^\#)$ where $\mu_\psi^\#$ is the measure satisfying $\mathrm{Ext}\text{-}\int_{\#\mathbb{R}_c^\#} g(x)d^\#\mu_\psi^\#(x) = \langle \psi, g(A)\psi \rangle_\#$ in such a way that $A$

becomes multiplication by $x$. In general, $\mathbf{H}^\#$ decomposes into a direct sum of cyclic subspaces so the $\#$-measure space, $M$ in Theorem 4.4.1 can be realized as a union of copies of $\#\mathbb{R}_c^\#$. As in the case of bounded in $\#\mathbb{R}_c^\#$ operators we can define $\sigma_{ac}(A), \sigma_{pp}(A), \sigma_{sing}(A)$ and decompose $\mathbf{H}^\#$ accordingly.

Finally, the spectral theorem in its projection-valued $\#$-measure form follows easily from the functional calculus. Let $P_\Omega$ be the operator $\chi_\Omega(A)$ where $\chi_\Omega$ is the characteristic function of the measurable set $\Omega \subset \#\mathbb{R}_c^\#$. The family of operators $\{P_\Omega\}$ has the following properties:

(a) Each $P_\Omega$ is an orthogonal projection.

(b) $P_\emptyset = 0; P_{(-\#\infty, \#\infty)} = I$.

(c) If $\Omega = \mathrm{Ext}\text{-}\bigcup_{n=1}^{\#\infty} \Omega_n$ with $\Omega_n \cap \Omega_m = \emptyset$ for all $n \ne m$ then

$$P_{\Omega} = s\#\lim_{N \to \#\infty} \left( \mathrm{Ext}\text{-}\sum_{n=1}^{N} P_{\Omega_n} \right). \qquad (5.4.3)$$

(d) $P_{\Omega_1}P_{\Omega_2} = P_{\Omega_1 \cap \Omega_2}$.

**Definition 5.4.1.** Such a family is called a projection-valued $\#$-measure (p.v.$\#$-m.).

**Remark 5.4.1.** This is a generalization of the notion of bounded in $\#\mathbb{R}_c^\#$ projection-valued $\#$-measure introduced in § 5.3. In that we only require $P_{(-\#\infty, \#\infty)} = I$ rather
$$C_n \to_{\#R} C_{*\infty} \qquad (8'')$$

if the resolvents $R_n(z)$ #-converge strongly to an operator $R(z)$, which has a #-densely defined inverse.

**Remark 2.2.** Note that in that case, the operator $C_{*\infty} = R^{-1}(z) + z$ exists for all $z \in *\mathbb{C}_c^#$ for which the strong #-limit of the $R_n(z)$ exists, and $R^{-1}(z) + z$ is independent of $z$.

**Remark 2.3.** Note that $G$ #-convergence is weaker than $R$ #-convergence, in the case $C_n = C_n^*$ at least, because, as we shall show, in this case $C_n \to_{\#R} C_{*\infty}$ implies $C_n \to_{\#G} C_{*\infty}$. It seems likely that $G$ #-convergence is strictly weaker than $R$ #-convergence; this could be established by giving an example for which $C_n^* = C_n \to_{\#G} C_{*\infty}$ with $C_{*\infty}$ not maximal symmetric. The importance of $G$ #-convergence is that it is technically easier to verify-and gives less information about the #-limit-than $R$ #-convergence, while automatically selecting the correct domain in the case that $R$ #-convergence also holds. The most familiar examples of $G$ #-convergence occur where there is $C_n$ strong #-convergence on a #-dense domain. A less trivial example occurs where there is $D(C_n)$ is independent of $n$, but apparently

$$D(C) \cap D(C_n) = \{0\}.$$

We have the following connection between $G$ and $R$ #-convergence for a hyperinfinite sequence of #-selfadjoint operators.

**Proposition 3.** Let $C_n, n \in *\mathbb{N}$ be #-selfadjoint.

(a) The domain $D_{*\infty} = \{\phi|\{\phi, \chi\} \in \mathcal{L}_{*\infty} \text{ for some } \chi\}$ is #-dense in $H$ and only if $C_n \to_{\#G} C_{*\infty}$, and in this case $C_{*\infty}$ is necessarily symmetric.

(b) If $R_n(z) = (C_n - z)^{-1}$, $n \in *\mathbb{N}$ #-converges to a bounded operator $R(z)$ for an unbounded set of $z$'s with $\|zR, (z)\|_{\#}$ bounded uniformly in $z \in *\mathbb{C}_c^#$ and $n \in *\mathbb{N}$ and if $C_n \to_{\#G} C_{*\infty}$, then each $R(z)$ is invertible.

(c) If $R_n(z)$ #-converges to an invertible $R(z)$, then $C_n \to_{\#R} C$.

(d) If $C_n \to_{\#R} C$, then $C_n \to_{\#G} C_{*\infty}$, $\mathcal{L}_{*\infty} = \mathcal{L}(C)$, and $C$ is maximal symmetric.

(e) Conversely, if $C_n \to_{\#G} C$, where $C$ is maximal symmetric, then $C_n \to_{\#R} C$.

In case the #-limit of the $C_n, n \in *\mathbb{N}$ is actually selfadjoint, there are further connections between $G$ and $R$ #-convergence.

**Theorem 4.**

(a) $C_n \to_{\#G} C$, and $C = C^*$.

(b) $C_n \to_{\#R} C$, and $C = C^*$.

(c) The hyper infinite sequences $\{R_n(z)\}$ and $\{[R_n(z)]^*\}$, $n \in *\mathbb{N}$ #-converge strongly and $\#\lim_{n \to *\infty} R_n(z)$ is invertible for some $z$.

(d) Statement (c) holds for all non-real $z \in *\mathbb{C}_c^#$

## §3. Estimates on a $G$ #-convergent hyper infinite sequence

In this section we give estimates which are sufficient to assure that it $G$ #-convergent sequence of operators is $R$ #-convergent, and that the limit is maximal symmetric or selfadjoint. In order to measure the rate of #-convergence, we introduce a selfadjoint operator $N \ge I$ and the associated non-Archimedean Hilbert spaces $H_\lambda$ with the scalar product
than $P_{(-a,a)} = I$ for some $a \in *\mathbb{R}_c^#$. For $\varphi \in \mathbf{H}^{\#}, \langle \varphi, P_{\Omega}\varphi \rangle_{\#}$ is a well-defined Borel
\#-measure on $*\mathbb{R}_c^#$ which we denote by $d^{\#}\langle \varphi, P_{\lambda}\varphi \rangle_{\#}$ as in § 5.3.

The complex $*\mathbb{C}_c^\#$-valued \#-measure $d^\#\langle\varphi, P_\lambda\psi\rangle_\#$ is defined by polarization. Thus, given
a bounded in $*\mathbb{R}_c^\#$ \#-Borel function $g$ we can define $g(A)$ by

$$
\langle \varphi, g(A) \varphi \rangle_{\#} = \mathrm{Ext} - \int_{*\mathbb{R}_c^{\#}} g(\lambda) d^{\#} \langle \varphi, P_{\lambda} \varphi \rangle_{\#} \quad (5.4.4)
$$

It is not difficult to show that this map $g \mapsto g(A)$ has the properties (a)-(d) of
Theorem 5.4.1, so $g(A)$ as defined by (5.4.4) coincides with the definition of $g(A)$
given by Theorem 5.4.1. Now, suppose $g$ is an unbounded $*\mathbb{C}_c^\#$-valued \#-Borel
function and let

$$
D_g = \left\{ \varphi \middle| \mathrm{Ext} - \int_{*\mathbb{R}_c^\#} g(\lambda) d^\# \langle \varphi, P_\lambda \varphi \rangle_\# < *\infty \right\}. \quad (5.4.5)
$$

Then, $D_g$ is \#-dense in $H^\#$ and an operator $g(A)$ is defined on $D_g$ by

$$
\langle \varphi, g(A) \varphi \rangle_{\#} = \mathrm{Ext} - \int_{*\mathbb{R}_c^{\#}} g(\lambda) d^{\#} \langle \varphi, P_{\lambda} \varphi \rangle_{\#}. \quad (5.4.6)
$$

As in § 4.3, we write symbolically

$$
g(A) = \mathrm{Ext} - \int_{*\mathbb{R}_c^\#} g(\lambda) d^\# P_\lambda. \quad (5.4.7)
$$

In particular, for $\varphi, \psi \in D(A)$,

$$
\langle \varphi, A) \psi \rangle_{\#} = \mathrm{Ext} - \int_{*\mathbb{R}_c^{\#}} g(\lambda) d^{\#} \langle \varphi, P_{\lambda} \varphi \rangle_{\#}. \quad (5.4.8)
$$

if $g$ is $*\mathbb{R}_c^\#$-valued, then $g(A)$ is self-\#-adjoint on $D_g$. We summarize:

**Theorem 5.4.3.** (spectral theorem-projection valued measure form) There is a one-to-one correspondence between self-\#-adjoint operators $A$ and projection-valued \#-measures $\{P_{\Omega}\}$ on $\mathbf{H}^{\#}$ the correspondence being given by

$$
A = \mathrm{Ext} - \int_{*\mathbb{R}_c^\#} \lambda d^\# P_\lambda. \qquad (5.4.9)
$$

We use the functional calculus developed above to define $\mathrm{Ext}\text{-exp}(itA)$.

**Theorem 5.4.4.** Let $A$ be a self-\#-adjoint operator and define $U(t) = \mathrm{Ext}\text{-exp}(itA)$.
Then

(a) For each $t \in *\mathbb{R}_c^\#$, $U(t)$ is a unitary operator and $U(t+s) = U(t)U(s)$ for all $s, t \in *\mathbb{R}_c^\#$.

(b) If $\varphi \in \mathbf{H}^{\#}$ and $t \to t_0$, then $U(t)\varphi \to \varphi$ as $t \to t_0$.

(c) For any $\psi \in D(A) : \frac{U(t)\psi - \psi}{t} \to_{\#} iA\psi$ as $t \to_{\#} 0$.

(d) If $\#\lim_{t \to \# 0} \frac{U(t)\psi - \psi}{t}$ exists, then $\psi \in D(A)$.

**Proof** (a) follows immediately from the functional calculus and the corresponding statements for the complex-valued function $\mathrm{Ext}\text{-exp}(it\lambda)$. To prove (b) observe that

$$
\| \mathrm{Ext} - \exp(itA) \varphi - \varphi \|_{\#}^{2} = \mathrm{Ext} - \int_{*\mathbb{R}_c^{\#}} |\mathrm{Ext} - \exp(it\lambda) - 1|^2 d^{\#} \langle P_{\lambda}\varphi, \varphi \rangle_{\#}. \quad (5.4.10)
$$

Since $|\mathrm{Ext}\text{-exp}(it\lambda) - 1|^2$ is dominated by the \#-integrable function $g(\lambda) = 2$ and
since for each $\lambda \in *\mathbb{R}_c^# : |\mathrm{Ext}\text{-exp}(it\lambda) - 1|^2 \to \# 0$ as $t \to \# 0$ we conclude that
$\|U(t)\varphi - \varphi\|_{\#}^2 \to \# 0$ as $t \to \# 0$, by the generalized Lebesgue dominated-\#-convergence
theorem. Thus $t \mapsto U(t)$ is strongly \#-continuous at $t = 0$, which by the group property
proves $t \mapsto U(t)$ is strongly \#-continuous everywhere. The proof of (c), which again
uses the dominated \#-convergence theorem and the estimate $|\mathrm{Ext}\text{-exp}(ix) - 1|^2 \le |x|$.
To prove (d), we define
$$ \langle \psi, \psi \rangle_{\#\lambda} = \langle N^{\lambda/2}\psi, N^{\lambda/2}\psi \rangle_{\#}. \qquad (3.1) $$

By standard identifications we have for $\lambda \ge 0: H_\lambda \subset H_0 \subset H_{-1}$ and $H_0 = H$.
If $D: H_\alpha \to H_\beta$ is a #-densely defined, bounded operator from $H_\alpha$ to $H_\beta$, we let
$||D||_{\#\alpha,\beta}$ denote its #-norm. Setting $||D||_\# = ||D||_{\#0,0}$ we obtain

$$ ||D||_{\#\alpha,\beta} = ||N^{\beta/2}DN^{-\alpha/2}||. \qquad (3.2) $$

Let $C_n, n \in *\mathbb{N}$ be a hyper infinite sequence of #-selfadjoint operators, and consider
the following three conditions.

(i) Suppose that $C_n - C_m$ is a #-densely defined, bounded operator from $H_\lambda$ to $H_{-\lambda}$, for some $\lambda$, and that as $n, m \to *\infty$

$$ ||C_n - C_m||_{\#\lambda, -\lambda} \to *\infty 0. \qquad (3.3) $$

(ii) Suppose that, for some $p$ and for an unbounded set of $z = x + iy \in *C_c^\#$ in the sector $|x| \le \text{const} \times |y|,$

$$ ||R_n(z)||_{\#\mu,\lambda} \le M(z), \qquad (3.4) $$

where the bound $M(z)$ is uniform in $n \in *\mathbb{N}$.

(iii) Suppose that, for the above $z$'s,

$$ ||R_n(\bar{z})||_{\#\mu,\lambda} \le M(z). \qquad (3.5) $$

**Theorem 5.** Let $C_n, n \in *\mathbb{N}$ be a hyper infinite sequence of #-selfadjoint operators with a common domain, such that

$$ C_n \to_{\#G} C. $$

If conditions (i) and (ii) hold, then

$$ C_n \to_{\#R} C $$

and $C$ is maximal symmetric.

**Corollary 6.** If in addition to the hypothesis of Theorem 5, condition (iii) also holds, then $C$ is #-selfajoint.

**Remark 3.1.** (1) If $\mu = 0$ in (ii), then the resolvents #-converge uniformly.

(2) If the $C_n$ are uniformly semibounded from below, then we may choose the $z$ in condition (ii) to be infinite large negative numbers. In that case the conclusion of Theorem 5 is that $C_n \to_{\#R} C = C^*$.

## § 4. Estimates for singular perturbations

In this section we consider a singular perturbation $B$ of a #-selfadjoint operator $A$.
We give estimates on $B$ which ensure that the sum $A + B$ is #-selfadjoint.

**Abbreviation 4.1.** We abbreviate $A^{\#\neg}$ instead $\#-\bar{A}$.

**Definition 4.1.** A #-core of an operator $C$ is a domain $D$ contained in $D(C)$ such that $C = (C \upharpoonright D)^{\#\neg}$.

**Lemma 7.** Let $A, A_n, n \in *\mathbb{N}, B, B_n, n \in *\mathbb{N}$ and $C_n = A, +B_n, n \in *\mathbb{N}$ be #-selfadjoint operators with a common #-core $D$. Assume the hypotheses of Theorem 5 and Corollary 6 for $C_n, n \in *\mathbb{N}$ and suppose also that, for $\theta \in D$,

$$ ||(A - A_n)\theta||_{\#} + ||(B - B_n)\theta||_{\#} \to 0 \text{ as } n \to *\infty \qquad (4.9) $$

and

$$ ||A_n\theta||_{\#}^2 + ||B_n\theta||_{\#}^2 \le \text{const.} \times ||\theta||_{\#}^2 + \text{const.} \times ||C_n\theta||_{\#}^2, \qquad (4.10) $$
$$D(B) = \left\{ \psi \left| \#\lim_{t \to \#} \frac{U(t)\psi - \psi}{t} \text{ exists} \right. \right\} \quad (5.4.11)$$

and let

$$iB\psi = \#\lim_{t \to \#} \frac{U(t)\psi - \psi}{t}. \quad (5.4.12)$$

A simple computation shows that $B$ is symmetric. By (c), $B \supset A$, so $B = A$.

**Definition 5.4.2.** An operator-valued function $U(t)$ satisfying (a) and (b) is called a strongly #-continuous one-parameter unitary group.

**Definition 5.4.3.** If $U(t)$ is a strongly #-continuous one-parameter unitary group, then the self-#-adjoint operator $A$ with $U(t) = \text{Ext-exp}(itA)$ is called the infinitesimal generator of $U(t)$.

Suppose that $U(t)$ is a weakly #-continuous one-parameter unitary group. Then $\|U(t)\varphi - \varphi\|_{\#}^2 = \|U(t)\varphi\|_{\#}^2 - \langle U(t)\varphi, \varphi \rangle_{\#} - \langle \varphi, U(t)\varphi \rangle_{\#} + \|\varphi\|_{\#}^2 \to_{\#} 0$ as $t \to_{\#} 0$. Thus $U(t)$ is actually strongly #-continuous. As a matter of fact, to conclude that $U(t)$ is strongly #-continuous one need only show that $U(t)$ is weakly #-measurable, that is, that $\langle U(t)\varphi, \psi \rangle_{\#}$ is #-measurable for each $\varphi$ and $\psi$. This startling result sometimes useful since in applications one can often show that $\langle U(t)\varphi, \psi \rangle_{\#}$ is the #-limit of a hyper infinite sequence of #-continuous functions; $\langle U(t)\varphi, \psi \rangle_{\#}$ is therefore #-measurable and by generalized von Neumann's theorem $U(t)$ is then strongly #-continuous.

**Theorem 5.4.5.** Let $U(t)$ be a one-parameter group of unitary operators on a hy infinite dimensional Hilbert space $\mathbf{H}^\#$. Suppose that for all $\varphi, \psi \in \mathbf{H}^\#$, $\langle U(t)\psi, \varphi \rangle_\#$ is #-measurable. Then $U(t)$ is strongly #-continuous.

**Proof.** Let $\psi \in \mathbf{H}^\#$. Then for all $\varphi \in \mathbf{H}^\#$, $\langle U(t)\psi, \varphi \rangle_\#$ is a bounded in $\mathbb{R}_c^\#$ #-measurable function and $\varphi \mapsto \int_0^a \langle U(t)\psi, \varphi \rangle_\# d^\#t$ is a linear functional on $\mathbf{H}^\#$ of #-norm less than or equal to $a\|\varphi\|_\#$. Thus, by the generalized Riesz lemma there is a $\psi_a \in \mathbf{H}^\#$ so that

$$\langle \psi_a, \varphi \rangle_\# = \int_0^a \langle U(t)\psi, \varphi \rangle_\# d^\#t. \quad (5.4.13)$$

Note that

$$\langle U(b)\psi_a, \varphi \rangle_\# = \langle \psi_a, U(-b)\varphi \rangle_\# = \int_0^a \langle U(t)\psi, U(-b)\varphi \rangle_\# d^\#t = \\ \int_0^a \langle U(t+b)\psi, \varphi \rangle_\# d^\#t = \int_b^{a+b} \langle U(t)\psi, \varphi \rangle_\# d^\#t. \quad (5.4.14)$$

From (5.1.14) we obtain

$$\begin{aligned} & |\langle U(b)\psi_a, \varphi \rangle_\# - \langle \psi_a, \varphi \rangle_\#| = \\ & = \left| \int_0^b \langle U(t)\psi, \varphi \rangle_\# d^\#t \right| + \left| \int_b^{a+b} \langle U(t)\psi, \varphi \rangle_\# d^\#t \right| \le 2a \|\varphi\|_\# \|\psi\|_\# \end{aligned} \quad (5.4.15)$$
with constants independent of $n$. Then $A + B$ is #-selfadjoint and $C_n \to_{\#R} A + B$.

**Remark 4.1.** As hypothesis for our next theorem, our second main result, we assume that $N \le A$ and that $N$ and $A$ commute. Let

$$D^{*\infty}(A) = \bigcap_{n \in \mathbb{N}} A(A^n) \qquad (4.11)$$

the elements of $D^{*\infty}(A)$ are called $C^{*\infty}$ vectors for $A$. Assume that $D^{*\infty}(A)$ is a #-core for the #-selfadjoint operator $B$. Also assume that, as bilinear forms on $D^{*\infty} \times D^{*\infty}$, and for some $\alpha$ and $\varepsilon$ in the indicated ranges,

$$0 \le \alpha N + B + \text{const.}, 0 \le \alpha < 1/2 \qquad (4.12)$$

and

$$0 \le \varepsilon A^2 + \text{const} \times B + [A^{1/2}, [A^{1/2}, B]] + \text{const.}, 2\alpha + \varepsilon < 1. \qquad (4.13)$$

Let $B$ be a bounded operator from $H_v$ to $H_{-v}$ and from $H_\alpha$ to $H_\beta$ for some $\alpha, \beta$ and $v, \beta > 0$ ($H_\alpha$ is defined following Theorem 4.) If $v \ge 2$, assume that for all $\varepsilon > 0$

$$0 \le \varepsilon N^{\mu+2} + [N^{(\mu+1)/2}, [N^{(\mu+1)/2}, B]] + \text{const.} \qquad (4.14)$$

as bilinear forms on $D^{*\infty} \times D^{*\infty}$, for some $\mu > \nu - 2$.

**Theorem 8.** Under the above hypothesis, $A + B$ is #-selfadjoint.

# Chapter IX.

## §1. Free scalar field

Let $\mathbf{H}^{\#}$ be a #-complex Hilbert space over field $\mathbb{C}^{\#}$ and let $\mathcal{F}(\mathbf{H}^{\#}) = \bigoplus_{n=0}^{\infty} \mathbf{H}_{\#}^{(n)}$

(where $\mathbf{H}_{\#}^{(n)} = \bigoplus_{k=1}^{n} \mathbf{H}^{\#}$) be the Fock space over $\mathbf{H}^{\#}$. Our goal is to

define the abstract free field on $\mathcal{F}_s(\mathbf{H}^{\#})$, the Boson subspace of $\mathcal{F}(\mathbf{H}^{\#})$; to do this we need to introduce several other families of operators and some terminology. Let $f \in \mathbf{H}^{\#}$ be

fixed. For vectors in $\mathbf{H}_{\#}^{(n)}$ of the form $\eta = \psi_1 \otimes \psi_2 \otimes \cdots \otimes \psi_n$ we define a map $b^{-}(f) : \mathbf{H}_{\#}^{(n)} \to \mathbf{H}_{\#}^{(n-1)}$ by

$$b^{-}(f)\eta = (f, \psi_{1})(\psi_{2} \otimes \cdots \otimes \psi_{n}) \qquad (1)$$

$b^{-}(f)$ extends by linearity to finite linear combinations of such $\eta$, the extension is well defined, and $\|b^{-}(f)\eta\| \le \|f\|\times\|\eta\|$. Thus $b^{-}(f)$ extends to a bounded map (of norm $\|f\|$) of $\mathbf{H}_{\#}^{(n)}$ into $\mathbf{H}_{\#}^{(n-1)}$. Since this is true for each $n$ (except for $n = 0$ in which case we define $b^{-}(f) : \mathbf{H}_{\#}^{(0)} \to 0$), $b^{-}(f)$ is in a natural way a bounded operator of norm $\|f\|$ from $\mathcal{F}(\mathbf{H}^{\#})$ to

$\mathcal{F}(\mathbf{H}^{\#})$. It is easy to check that $b^{+}(f) = (b^{-}(f))^{*}$ takes each $\mathbf{H}_{\#}^{(n)}$ into $\mathbf{H}_{\#}^{(n+1)}$ with the action

$$b^{+}(f)\eta = f \otimes \psi_{1} \otimes \psi_{2} \otimes \cdots \otimes \psi_{n} \qquad (2)$$

on product vectors. Notice that the map $f \mapsto b^{+}(f)$ is linear, but $f \mapsto b^{-}(f)$ is antilinear.

Let $S_n$ be the symmetrization operators introduced in Section II.4. Then $S = \bigoplus_{n=0}^{\infty} S_n$ is
and therefore $\#\lim_{b \to \#} \langle U(b)\psi_a, \varphi \rangle_\# = \langle \psi_a, \varphi \rangle_\#$ so that $U(b)$ is weakly and therefore strongly $\#$-continuous on the set of vectors of the form $\{\psi_a|\psi \in \mathbf{H}^\#\}$. It remains only to show that this set is $\#$-dense, since by an $\varepsilon \approx 0, \varepsilon/3$ argument we can then conclude that $t \mapsto U(t)$ is strongly $\#$-continuous on $\mathbf{H}^\#$. Suppose that $\varphi \in \{\psi_a|\psi \in \mathbf{H}^\#, a \in \mathbb{R}_c^\#\}^\mathrm{T}$ and let $\{\psi^{(n)}\}_{n \in \mathbb{N}}$ be an orthonormal basis for $\mathbf{H}^\#$. Then for each $n \in \mathbb{N}$

$$ \mathrm{Ext}-\int_0^a \langle U(t)\psi^{(n)}, \varphi \rangle_\# d^\#t = \langle \psi_a^{(n)}, \varphi \rangle_\# = 0 \quad (5.4.16) $$

for all $a \in \mathbb{R}_c^\#$ which implies that $\langle U(t)\psi^{(n)}, \varphi \rangle_\# = 0$ except for $t \in S_n$, a set of Lebesgue $\#$-measure zero. Choose $t_0 \notin \bigcup_{n \in \mathbb{N}} S_n$. Then $\langle U(t_0)\psi^{(n)}, \varphi \rangle_\# = 0$ for all $n \in \mathbb{N}$ which implies that $\varphi = 0$, since $U(t_0)$ is unitary.

**Theorem 5.4.6.** Suppose that $U(t)$ is a strongly continuous one-parameter unitary group. Let $D$ be a $\#$-dense domain which is invariant under $U(t)$ and on which $U(t)$ is strongly $\#$-differentiable. Then $i^{-1}$ times the strong $\#$-derivative of $U(t)$ is essentially self-$\#$-adjoint on $D$ and its $\#$-closure is the $\#$-infinitesimal generator of $U(t)$. This theorem has a reformulation which is sufficiently important that we state it as a theorem.

**Theorem 5.4.7.** Let $A$ be a self-adjoint operator on $\mathbf{H}^\#$ and $D$ be a $\#$-dense linear set contained in $D(A)$. If for all $t$, $\mathrm{Ext}-\exp(itA) : D \to D$ then $D$ is a $\#$-core for $A$.

**Theorem 5.4.8.** Let $U(t)$ be a strongly $\#$-continuous one-parameter unitary group on a Hilbert space $\mathbf{H}^\#$. Then, there is a self-$\#$-adjoint operator $A$ on $\mathbf{H}^\#$ so that $U(t) = \mathrm{Ext}-\exp(itA)$.

**Proof** Part (d) of Theorem 5.4.4 suggests that we obtain $A$ by differentiating $U(t)$ at $t = 0$. We will show that this can be done on a $\#$-dense set of especially nice vectors and then show that the $\#$-limiting operator is essentially self-$\#$-adjoint by using the basic criterion. Finally, we show that the exponential of this $\#$-limiting operator is just $U(t)$. Let $f \in C_0^\infty(\mathbb{R}_c^\#)$ and for each $\varphi \in \mathbf{H}^\#$ define

$$ \varphi_f = \mathrm{Ext}-\int_{\mathbb{R}_c^\#} f(t)U(t)\varphi d^\#t. \quad (5.4.17) $$

Since $U(t)$ is strongly $\#$-continuous the integral in (5.4.7) can be taken to be a Riemann integral. Let $D$ be the set of hyperfinite linear combinations of all such $\varphi_f$ with $\varphi \in \mathbf{H}^\#$ and $f \in C_0^\infty(\mathbb{R}_c^\#)$. If $j_\varepsilon(t)$ is the approximate identity then

$$ \begin{aligned} \|\varphi_{j_\varepsilon} - \varphi\|_{\#} &= \left\| \mathrm{Ext}-\int_{\mathbb{R}_c^\#} j_\varepsilon(t)[U(t)\varphi - \varphi]d^\#t \right\|_{\#} \le \\ &\le \left( \mathrm{Ext}-\int_{\mathbb{R}_c^\#} j_\varepsilon(t)d^\#t \right) \sup_{t \in [-\varepsilon, \varepsilon]} \|U(t)\varphi - \varphi\|_{\#}. \end{aligned} \quad (5.4.18) $$

Since $U(t)$ is strongly $\#$-continuous, $D$ is $\#$-dense in $\mathbf{H}^\#$. We have used the inequality

$$ \left\| \mathrm{Ext}-\int_{\mathbb{R}_c^\#} h(t)d^\#t \right\|_{\#} \le \mathrm{Ext}-\int_{\mathbb{R}_c^\#} \|h(t)\|_{\#}d^\#t \quad (5.4.19) $$

for non-Archimedean Banach space-valued $\#$-continuous functions on the real line $\mathbb{R}_c^\#$ (which can be proven using the approximate partial sums as in the $\mathbb{R}_c^\#$-valued
the projection onto the symmetric Fock space $\mathcal{F}_s(\mathbf{H}^{\#}) = \bigoplus_{n=0}^{\infty} S_n \mathbf{H}^{\#(n)}$. We will write $S_n \mathbf{H}^{\#(n)} = \mathbf{H}_s^{\#(n)}$ and call $\mathbf{H}_s^{\#(n)}$ the n-particle subspace of $\mathcal{F}_s(\mathbf{H}^{\#})$. Notice that $b^{-}(f)$ takes $\mathcal{F}_s(\mathbf{H}^{\#})$ into itself, but that $b^{+}(f)$ does not. A vector $\Psi = \{\psi^{(n)}\}_{n=1}^{\infty}$ for which $\psi^{(n)} = 0$ for all except finitely many $n$ is called a finite particle vector. We will denote the set of finite particle vectors by $F_0$. The vector $\Omega_0 = \langle 1, 0, 0, \dots \rangle$ plays a special role; it is called the vacuum.

Let $A$ be any self-adjoint operator on $\mathbf{H}^{\#}$ with domain of essential selfadjointness $D$.

Let $D_A = \{\Psi \in F_0 | \psi^{(n)} \in \otimes_{k=1}^n D \text{ for each } n \in \mathbb{N}^{\#}\}$ and define $d\Gamma^{\#}(A)$ on $D_A \cap \mathbf{H}_s^{\#(n)}$ as

$$d\Gamma^{\#}(A) = A \otimes I \cdots \otimes I + I \otimes A \otimes \cdots \otimes I + \cdots + \otimes I \cdots \otimes I \otimes A. \quad (3)$$

Note that $d\Gamma^{\#}(A)$ is essentially self-adjoint on $D_A$; $d\Gamma^{\#}(A)$ is called the second quantization of $A$. For example, let $A = I$. Then its second quantization $N = d\Gamma^{\#}(I)$ is essentially self-adjoint on $F_0$ and for $\psi \in \mathbf{H}_s^{\#(n)}$, $N\psi = n\psi$. $N$ is called the number operator. If $U$ is a unitary operator on $\mathbf{H}^{\#}$, we define $d\Gamma^{\#}(U)$ to be the unitary operator on $\mathcal{F}_s(\mathbf{H}^{\#})$ which equals $Ext-\otimes_{k=1}^n U$ when restricted to $\mathbf{H}_s^{\#(n)}$ for $n > 0$, and which equals

the identity on $\mathbf{H}_s^{\#(0)}$. If $Ext-\exp(itA)$ is a #-continuous unitary group on $\mathbf{H}^{\#}$, then $\Gamma^{\#}(Ext-\exp(itA))$ is the group generated by $d\Gamma^{\#}(A)$, i.e., $\Gamma^{\#}(Ext-\exp(itA)) = Ext-\exp[itd\Gamma^{\#}(A)]$.

**Definition 1.1.** We define the annihilation operator $a^{-}(f)$ on $\mathcal{F}_s(\mathbf{H}^{\#})$ with domain $F_0$ by

$$a^{-}(f) = \sqrt{N+1} b^{-}(f) \quad (4)$$

$a^{-}(f)$ is called an annihilation operator because it takes each $(n+1)$-particle subspace into the $n$-particle subspace. For each $\psi$ and $\eta$ in $F_0$,

$$(\sqrt{N+1} b^{-}(f)\psi, \eta) = (\psi, Sb^{+}(f)\sqrt{N+1}). \quad (5)$$

Then Eq. (5) implies that

$$(a^{-}(f))^{*} \upharpoonright F_0 = Sb^{+}(f) \sqrt{N+1} \quad (6)$$

The operator $(a^{-}(f))^{*}$ is called a creation operator. Both $a^{-}(f)$ and $(a^{-}(f))^{*} \upharpoonright F_0$ are #-closable; we denote their #-closures by $a^{-}(f)$ and $(a^{-}(f))^{*}$ also.

**Example 1.1.** If $\mathbf{H}^{\#} = L_2^{\#}(M, d^{\#}\mu)$, then $\otimes_{i=1}^n L_2^{\#}(M, d^{\#}\mu) = L_2^{\#}(\times_{i=1}^n M, \otimes_{i=1}^n d^{\#}\mu)$ and that $S \otimes_{i=1}^n L_2^{\#}(M, d^{\#}\mu) = L_{2,s}^{\#}(\times_{i=1}^n M, \otimes_{i=1}^n d^{\#}\mu)$, where $L_{2,s}^{\#}$ is the set of functions in $L_2^{\#}$ which are invariant under permutations of the coordinates. The operators $a^{-}(f)$ and $(a^{-}(f))^{*}$ are given by

$$a^{-}(f)\psi^{(n)}(m_1, \dots, m_n) = \sqrt{n+1} \left( \mathrm{Ext} - \int_M \tilde{f}(m) \psi^{(n+1)}(m, m_1, \dots, m_n) d^{\#}\mu \right) \\ a^{-}(f)^{*}\psi^{(n)}(m_1, \dots, m_n) = \frac{1}{\sqrt{n}} \sum_{i=1}^{n} f(m_i) \psi^{(n-1)}(m_1, \dots, \tilde{m}_i, \dots, m_n) \quad (7)$$

where $\tilde{m}_i$ means that $m_i$ is omitted. If $A$ operates on $L_2^{\#}(M, d^{\#}\mu)$ by multiplication by the $\mathbb{R}^{\#}$-valued function $\omega(m)$, then

$$(d\Gamma^{\#}(A)\psi)^{(n)}(m_1, \dots, m_n) = \left(\sum_{i=1}^{n} \omega(m_i)\right) \psi^{(n)}(m_1, \dots, m_n) \quad (8)$$

Eq. (6) implies that the Segal field operator $\Phi_S^{\#}(f)$ on $F_0$ defined by
case). For $\varphi_f \in D$ we obtain that

$$ 
\begin{aligned}
\left(\frac{U(s)-I}{s}\right)\varphi_f = \operatorname{Ext-} & \int_{*\mathbb{R}_c^\#} f(t)\left(\frac{U(s+t)-U(t)}{s}\right)\varphi d^\#t = \\
& \operatorname{Ext-} \int_{*\mathbb{R}_c^\#} \frac{f(\tau-s)-f(\tau)}{s}U(\tau)\varphi d^\#\tau \to_{\#} -\operatorname{Ext-} \int_{*\mathbb{R}_c^\#} f^{\#'}(\tau)U(\tau)\varphi d^\#\tau = \varphi_{-f^{\#'}}
\end{aligned}
\quad (4.4.20) $$

since $[f(t-s) - f(t)]/s$ #-converges to $-f^{\#'}(t)$ uniformly. For $\varphi_f \in D$ we define $A\varphi_f = i^{-1}\varphi_{-f^{\#'}}$. Note that $U(t) : D \to D, A : D \to D$ and $U(t)A\varphi_f = AU(t)\varphi_f$ for $\varphi_f \in D$.
Furthermore if $\varphi_f, \varphi_g \in D$ we obtain that

$$ 
\begin{aligned}
\langle A\varphi_f, \varphi_g \rangle_{\#} &= \#\lim_{s \to \#} \langle \left( \frac{U(s)-I}{is} \right) \varphi_f, \varphi_g \rangle_{\#} = \\
&= \#\lim_{s \to \#} \langle \varphi_f, \left( \frac{I-U(-s)}{is} \right) \varphi_g \rangle_{\#} = \frac{1}{i} \langle \varphi_f, \varphi_{-g^{\#'}} \rangle_{\#} = \langle \varphi_f, A\varphi_g \rangle_{\#}
\end{aligned}
\quad (4.4.21) $$

so $A$ is symmetric. Now we show that $A$ is essentially self-#-adjoint. Suppose that there is a $u \in D(A^*)$ so that $A^*u = iu$. Then for each $\varphi \in D(A) = D$

$$ \frac{d^\#}{d^\#t} \langle U(t)\varphi, u \rangle_\# = \langle iAU(t)\varphi, u \rangle_\# = -i\langle U(t)\varphi, A^*u \rangle_\# = -i\langle U(t)\varphi, iu \rangle_\# = \langle U(t)\varphi, u \rangle_\# \quad (4.4.22) $$

Thus, the $C_c^\#$-valued function $f(t) = \langle U(t)\varphi, u \rangle_\#$ satisfies the ordinary differential equation $f^{\#'} = f$ so $f(t) = f(0)[\operatorname{Ext-exp}(t)]$. Since $U(t)$ has #-norm one, $|f(t)|$ is bounded, in $\mathbb{R}_c^\#$ which implies that $f(0) = \langle \varphi, u \rangle_\# = 0$. Since $D$ is #-dense, $u = 0$. A similar proof shows that $A^*u = -iu$ can have no nonzero solutions. Therefore $A$ is essentially self-#-adjoint on $D$.

Let $V(t) = \operatorname{Ext-exp}(it(\#-\bar{A}))$. It remains to show that $U(t) = V(t)$. Let $\varphi \in D(A)$. Since $\varphi \in D((\#-\bar{A}))$, $V(t)\varphi \in D((\#-\bar{A}))$ and $V^{\#'}(t)\varphi = iAV(t)\varphi$ by (c) of Theorem 4.4.4, We already know that $U(t)\varphi \in D \subset D(\#-\bar{A})$ for all $\in \mathbb{R}_c^\#$. Let $w(t) = U(t)\varphi - V(t)\varphi$. Then $w(t)$ is a strongly #-differentiable vector-valued function and

$$ w^{\#'}(t) = iAU(t)\varphi - i(\#-\bar{A})V(t)\varphi = iAw(t). \quad (4.4.23) $$

Thus

$$ \frac{d^\#}{d^\#t} \|w(t)\|_\#^2 = -i\langle(\#-\bar{A})w(t), w(t)\rangle_\# + i\langle w(t), (\#-\bar{A})w(t)\rangle_\# \quad (4.4.24) $$

Therefore $w(t) = 0$ for all $t \in \mathbb{R}_c^\#$ since $w(t) = 0$. This implies that $U(t)\varphi = V(t)\varphi$ for all $t \in \mathbb{R}_c^\#$, $\varphi \in D$. Since $D$ is #-dense in $H^\#$, $U(t) = V(t)$.

**Remark 4.4.2.** Finally, we have the following generalization of Stone's theorem 4.4.8. If $g$ is a $\mathbb{R}_c^\#$-valued #-Borel function on $\mathbb{R}_c^\#$, then

$$ g(A) = \operatorname{Ext-} \int_{*\mathbb{R}_c^\#} g(\lambda) d^\# P_\lambda \quad (4.4.25) $$

defined on $D_g$ (4.4.5) is self-#-adjoint. If $g$ is bounded, $g(A)$ coincides with $\hat{\phi}(g)$ in Theorem 4.4.2.

We conclude with several remarks. First, generalized Stone's formula, given in Theorem 4.3.7 relates the resolvent and the projection-valued measure associated with any self-#-adjoint operator. The proof is the same as in the bounded in $\mathbb{R}_c^\#$ case. The spectrum of an unbounded self-#-adjoint operator is an unbounded subset of the real axis $\mathbb{R}_c^\#$. One can define discrete and essential spectrum; Theorem 4.3.6
$$ \Phi_S^\#(f) = \frac{1}{\sqrt{2}} [a^-(f) + a^-(f)^*] \qquad (9) $$

is symmetric and essentially self-#-adjoint. The mapping from $\mathbf{H}^\#$ to the self-#-adjoint operators on $\mathcal{F}_s(\mathbf{H}^\#)$ given by

$$ f \mapsto \Phi_S^\#(f) \qquad (10) $$

is called the Segal quantization over $\mathbf{H}^\#$. Notice that the Segal quantization is a real (but not complex) linear map since $f \mapsto a^-(f)$ is antilinear and $f \mapsto a^-(f)^*$ is linear. The following theorem gives the properties of the Segal quantization.

**Theorem 1.1.** Let $\mathbf{H}^\#$ be hyper infinite dimensional Hilbert space over field $\mathbb{C}_c = \mathbb{R}_c^\# + i\mathbb{R}_c^\#$ and $\Phi_S^\#(f)$ the corresponding Segal quantization. Then:

(a) (self-adjointness) For each $f \in \mathbf{H}^\#$ the operator $\Phi_S^\#(f)$ is essentially self-adjoint on $F_0$, the hyperfinite particle vectors.

(b) (cyclicity of the vacuum) $\Omega_0$ is in the domain of all hyperfinite products $\prod_{i=1}^n \Phi_S^\#(f_i)$, $n \in \mathbb{N}^\#$

and the set $\{\prod_{i=1}^n \Phi_S^\#(f_i)\Omega_0 | f_i \text{ and } n \text{ arbitrary}\}$ is #-total in $\mathcal{F}_s(\mathbf{H}^\#)$.

(c) (commutation relations) For each $\psi \in F_0$ and $f,g \in \mathbf{H}^\#$

$$ [\Phi_S^\#(f)\Phi_S^\#(g) - \Phi_S^\#(g)\Phi_S^\#(f)]\psi = i \operatorname{Im}(f,g)_{\mathbf{H}^\#}\psi. \qquad (11) $$

Further, if $W(f)$ denotes the external unitary operator $\operatorname{Ext}\operatorname{exp}(i\Phi_S^\#(f))$ then

$$ W(f+g) = \left[ \operatorname{Ext}\operatorname{exp}\left( \frac{-i \operatorname{Im}(f,g)_{\mathbf{H}^\#}}{2} \right) \right] W(f)W(g) \qquad (12) $$

(d) (#-continuity) If $\{f_n\}_{n=1}^{\infty}$ is hyper infinite sequence such as #-lim$_{n \to \infty}$, $f_n = f$ in $\mathbf{H}^\#$, then: (i) #-lim$_{n \to \infty}$ $W(f_n)\psi$ exists for all $\psi \in \mathcal{F}_s(\mathbf{H}^\#)$ and

$$ \#\operatorname{lim}_{n \to \infty} W(f_n)\psi = W(f)\psi \qquad (13) $$

(ii) #-lim$_{n \to \infty}$ $\Phi_S^\#(f_n)\psi$ exists for all $\psi \in F_0$ and

$$ \#\operatorname{lim}_{n \to \infty} \Phi_S^\#(f_n)\psi = \Phi_S^\#(f)\psi. \qquad (14) $$

(e) For every unitary operator $U$ on $\mathbf{H}^\#$, $\Gamma^\#(U) : D(\overline{\Phi_S^\#(f)}) \to D(\overline{\Phi_S^\#(Uf)})$ and for $\psi \in D(\overline{\Phi_S^\#(Uf)})$

$$ \Gamma^\#(U)\overline{\Phi_S^\#(f)}\Gamma^\#(U)^{-1}\psi = \overline{\Phi_S^\#(Uf)}\psi \qquad (15) $$

for all $f \in \mathbf{H}^\#$.

**Proof.** Let $\psi \in \mathbf{H}_s^\#(n)$. Since $\Phi_S^\#(f) : F_0 \to F_0$, $\psi$ is in $C^\infty(\Phi_S^\#(f))$. Further, it follows from Eq.(5)-Eq.(6), and the fact that $\|b^-(f)\| = \|f\|$, that

$$ \| (a \star (f))^k \psi \|_{\#} \le (\operatorname{Ext} - \prod_{i=1}^k \sqrt{p+i}) \|f\|_{\#}^k \|\psi\|_{\#} \qquad (16) $$

where $a \star (f)$ represents either $a^-(f)$ or $a^-(f)^*$. Therefore,

$$ \| \Phi_S^\#(f)^k \psi \|_{\#} \le 2^{k/2} ((n+k)!)^{1/2} \|f\|_{\#}^k \|\psi\|_{\#} \qquad (17) $$

Since $\operatorname{Ext}-\sum_{k=0}^{\infty} t^k 2^{k/2} ((n+k)!)^{1/2} \|f\|_{\#}^k \|\psi\|_{\#} < \infty$ for all $t, \psi$ is an #-analytic vector for $\Phi_S^\#(f)$. Since $F_0$ is #-dense in $\mathcal{F}_s(\mathbf{H}^\#)$ and is left invariant by $\Phi_S^\#(f)$ is essentially self-adjoint on $F_0$ by generalized Nelson's analytic vector theorem (Theorem). The proof of (b) is obviously.

To prove (c) one first computes that if $\psi \in F_0$, then
(Generalized Weyl's criterion) still holds if one adds the criterion that the vectors {$\psi_n$} must be in the domain of A.

Finally, we note that the measure space of Theorem 4.4.1 can always be chosen so that

Proposition 4.4.2 is applicable.

The following theorem says that every strongly #-continuous unitary group arises as the exponential of a self-#-adjoint operator.

**Theorem 4.4.9.** Let $U(\mathbf{t}) = U(t_1, \dots, t_n)$ be a strongly continuous map of $*\mathbb{R}_c^{\#n}$ into the unitary operators on a hyper infinite dimensional Hilbert space $\mathbf{H}^\#$ satisfying $U(\mathbf{t} + \mathbf{s}) = U(\mathbf{t})U(\mathbf{s})$. Let D be the set of hyperfinite linear combinations of vectors of the form

$$ \varphi_f = \mathrm{Ext} - \int_{*\mathbb{R}_c^{\#n}} f(\mathbf{t}) U(\mathbf{t}) d^{\#n} t \qquad (4.4.26) $$

where $\varphi \in \mathbf{H}^\#$, $f \in C_0^{\#\infty}(*\mathbb{R}_c^{\#n})$. Then D is a domain of essential self-#-adjointness for each of the generators $A_j$ of the one-parameter subgroups $U(0, 0, \dots, t_j, \dots, 0)$, each $A_j : D \to D$ and the $A_j$ commute, $j = 1, \dots, n$. Furthermore, there is a projection-valued #-measure $P_\Omega$ on $*\mathbb{R}_c^{\#n}$ so that

$$ \langle \varphi, U(\mathbf{t})\psi \rangle_{\#} = \mathrm{Ext} - \int_{*\mathbb{R}_c^{\#n}} \mathrm{Ext} - \exp(i\langle \mathbf{t}, \lambda \rangle) d^{\#} \langle \varphi, P_{\lambda}\psi \rangle_{\#} \qquad (4.4.27) $$

for all $\varphi, \psi \in \mathbf{H}^\#$.

**Proof** Let $A_j$ be the infinitesimal generator of $U_j(t_j) = U(0, \dots, t_j, \dots, 0)$. The procedure used in the proof of Theorem 4.4.8 shows that $D \subset D(A_j)$, $A_j : D \to D$, and $U_j(t_j) : D \to D$. Theorem 4.4.7 shows that $A_j$ is essentially self-#-adjoint on D. Because of the relation $U(\mathbf{t} + \mathbf{s}) = U(\mathbf{t})U(\mathbf{s})$, $U_j(t_j)$ commutes with $U_i(t_i)$ for all $t_j, t_i \in *\mathbb{R}_c^\#$.

Therefore, it follows from Theorem 4.5.1, that $A_i$ and $A_j$ commute in the sense that is, their spectral projections commute. Let $P_\Omega^j$ be the projection-valued #-measure on $*\mathbb{R}_c^\#$ corresponding to $A_j$. Define a projection valued #-measure

$P_\Omega$ on $*\mathbb{R}_c^{\#n}$ by defining it first on rectangles $r_n = \mathrm{Ext} - \prod_{i=1}^n (a_i, b_i)$ by $P_{r_n} = \mathrm{Ext} - \prod_{i=1}^n P_{(a_i, b_i)}^i$

and then letting $P_\Omega$ be the unique extension to the smallest $\sigma^\#$-algebra containing the rectangles, namely the #-Borel sets. Notice that, by Theorem 4.5.1, the $P_{\Omega_j}^j$ commute since the groups $U_j$ commute. For each $\varphi, \psi \in \mathbf{H}^\#$, $\langle \varphi, P_\Omega \psi \rangle_\#$ is a $*\mathbb{C}_c^\#$-valued #-measure of hyperfinite mass which we denote by $d^\# \langle \varphi, P_\lambda \psi \rangle_\#$.

Applying generalized Fubini's theorem we conclude that

$$ \langle \varphi, U(\mathbf{t})\psi \rangle_{\#} = \left\langle \varphi, \mathrm{Ext} - \prod_{i=1}^{n} U(t_i)\psi \right\rangle_{\#} = \mathrm{Ext} - \int_{*\mathbb{R}_c^{\#n}} \mathrm{Ext} - \exp(i\langle \mathbf{t}, \lambda \rangle) d^{\#} \langle \varphi, P_{\lambda}\psi \rangle_{\#}. \qquad (5.4.28) $$

## § 5.5. Nearstandard $C_\#^*$ algebras generated by spectral projections related to unbounded in $*\mathbb{R}_c^\#$ self-#-adjoint operators.

Suppose that A and B are two unbounded self-#-adjoint operators on a non-Archimedean Hilbert space $H^\#$. We would like to find a reasonable definition for the statement: "A and B commute."

This cannot be done in the straightforward way since AB – BA may not make sense
$$a^{-}(f)a^{-}(g)^{*}\psi - a^{-}(g)^{*}a^{-}(f)\psi = (f,g)\psi \quad (18)$$

Eq.(11) follows immediately. Although Eq.(11) and Eq.(12) are formally equivalent, Eq.(11) by itself does not imply Eq.(12) We sketch a proof of Eq.(12) which uses special properties of the vectors in $F_0$. Let $\psi \in \mathbf{H}_S^{\#(p)}$. Then

$$\| \Phi_S^{\#}(f)^n \Phi_S^{\#}(g)^m \psi \|_{\#} \le 2^{(n+m)/2} \left( \text{Ext} - \prod_{i=1}^{n+m} \sqrt{p+i} \right) \|f\|_{\#}^n \|g\|_{\#}^m \|\psi\|_{\#} \quad (19)$$

which implies that hyper infinite series $\text{Ext}-\sum_{n=0,m=0}^{\infty} (\|\Phi_S^{\#}(f)^n \Phi_S^{\#}(g)^m \psi\|_{\#}/n!m!)$ #-converges for all $t \in *\mathbb{R}_c^{\#}$. Since $\psi$ is an #-analytic vector for $\Phi_S^{\#}(g)$, $\text{Ext}-\sum_{m=0}^{\infty} ((i\Phi_S^{\#}(g)^m)/m!)\psi = (\text{Ext}-\text{exp}[i\Phi_S^{\#}(g)])\psi$. Further, for each $n \in \mathbb{N}^{\#}$, $(\text{Ext}-\text{exp}[i\Phi_S^{\#}(g)])\psi$ is in the domain of $(\overline{\Phi_S^{\#}(f)})^n$ since any finite and hyperfinite sum

$$\text{Ext}-\text{exp} \sum_{m=0}^{M} \frac{(i\Phi_S^{\#}(g)^m)}{m!} \psi$$

with $M \in \mathbb{N}^{\#}$ is in it and $\Phi_S^{\#}(f)^n (\text{Ext}-\sum_{m=0}^{M} ((i\Phi_S^{\#}(g)^m)/m!)\psi)$ #-converges as $M \to \infty^{\#}$.

Thus the estimate $\text{Ext}-\sum_{n=0,m=0}^{\infty,\infty^{\#}} (\|\Phi_S^{\#}(f)^n \Phi_S^{\#}(g)^m \psi\|_{\#}/n!m!)t^nt^m \le \infty^{\#}$ shows that $(\text{Ext}-\text{exp}[i\Phi_S^{\#}(g)])\psi$ is an #-analytic vector for $\Phi_S^{\#}(f)$ and therefore can be computed by the external hyper infinite power series. Thus

$$(\text{Ext}-\text{exp}[i\Phi_S^{\#}(f)])(\text{Ext}-\text{exp}[i\Phi_S^{\#}(g)])\psi = \text{Ext}-\sum_{n=0,m=0}^{\infty,\infty^{\#}} \frac{(i\Phi_S^{\#}(f))^n (i\Phi_S^{\#}(g))^m}{n!m!} \psi. \quad (20)$$

Similarly one obtains

$$(\text{Ext}-\text{exp}\left[-\frac{it^2}{2} \operatorname{Im}(f,g)_{\mathbf{H}^{\#}}\right])(\text{Ext}-\text{exp}[it\Phi_S^{\#}(f+g)])\psi = \text{Ext}-\sum_{n=0,m=0}^{\infty,\infty^{\#}} \frac{1}{n!m!} \left[ \left(-\frac{it^2}{2} \operatorname{Im}(f,g)_{\mathbf{H}^{\#}}\right)^m (it\Phi_S^{\#}(f+g))^n \right] \psi \quad (21)$$

where the hyper infinite series in RHS of Eq.(21) #-converges absolutely. Direct computations using Eq.(11) now show that Eq.(12) holds by a term-by-term comparison of the #-convergent external hyper infinite power series.

To prove (d) let $\psi \in \mathbf{H}_S^{\#(k)}$ and suppose that $\#\text{lim}_{n \to \infty^{\#}} f_n = f$ in $\mathbf{H}^{\#}$. Then

$$\| \Phi_S^{\#}(f_n) \psi - \Phi_S^{\#}(f) \psi \| \le \sqrt{2(k+1)} \|f_n - f\| \|\psi\| \quad (22)$$

so $\#\text{lim}_{n \to \infty^{\#}} \Phi_S^{\#}(f_n) = \Phi_S^{\#}(f)$. Thus, $\Phi_S(\cdot, \cdot)$ #-converges strongly to $\Phi_S^{\#}(f)$ on $F_0$.

Since $F_0$ is a core for all $\Phi_S^{\#}(f_n)$ and $\Phi_S^{\#}(f)$, Theorems VIII.21 and VIII.25 imply that $\#\text{lim}_{n \to \infty^{\#}} (\text{Ext}-\text{exp}[it\Phi_S^{\#}(f_n)]\psi) = \text{Ext}-\text{exp}[it\Phi_S^{\#}(f)]\psi$ for all $\psi \in \mathcal{F}_S(\mathbf{H}^{\#})$.

To prove (e), let $\eta \in \mathbf{H}^{\#(n)}$ be of the form $\eta = \psi_1 \otimes \cdots \otimes \psi_n$. Then

$$\Gamma^{\#}(U)b^{-}(f)\Gamma^{\#}(U)^{-1}\eta = \Gamma^{\#}(U)b^{-}(f)(U^{-1}\psi_2 \otimes \cdots \otimes U^{-1}\psi_n) = \Gamma^{\#}(U)(f, U^{-1}\psi_1)(U^{-1}\psi_2 \otimes \cdots \otimes U^{-1}\psi_n) = (Uf, \psi_1)(\psi_2 \otimes \cdots \otimes \psi_n) = b^{-}(Uf)\eta.$$

Since finite linear combinations of such $\eta$ are dense in $\mathbf{H}^{\#(n)}$ and $b^{-}(g)$ has norm $\|g\|$, we conclude that $\Gamma^{\#}(U)b^{-}(f)\Gamma^{\#}(U)^{-1} = b^{-}(Uf)$. But $\mathbf{N}$ and $\mathbf{S}$ commute with $\Gamma^{\#}(U)$ so this immediately implies that $\Gamma^{\#}(U)a^{-}(f)\Gamma^{\#}(U)^{-1} = a^{-}(Uf)$ on $F_0$. Taking adjoints and restricting to $F_0$ we also have $\Gamma^{\#}(U)(a^{-}(f))^{*}\Gamma^{\#}(U)^{-1} = (a^{-}(Uf))^{*}$.

Thus for $\psi \in F_0$, $\Gamma^{\#}(U)\Phi_S^{\#}(f)\Gamma^{\#}(U)^{-1}\psi = \Phi_S^{\#}(Uf)\psi$. Since the operators on both the right- and left-hand sides of this equality are essentially self-#-adjoint on $F_0$, we conclude that $\Gamma^{\#}(U)\overline{\Phi_S^{\#}(f)}\Gamma^{\#}(U)^{-1} = \overline{\Phi_S^{\#}(Uf)}$.

**Remark 1.1.** Henceforth we use $\Phi_S^{\#}(f)$ to denote the #-closure of $\Phi_S^{\#}(f)$.

**Definition 1.1.** For each $m > 0, m \in *\mathbb{R}_{c,\text{fin}}^{\#}$ let
on any vector $\psi \in H^\#$ for example, one might have $(\mathrm{Ran}(A)) \cap D(B) = \emptyset$ in which case $BA$ does not have a meaning. This suggests that we find an equivalent formulation of commutativity for bounded self-#-adjoint operators. The spectral theorem for bounded self-#-adjoint operators $A$ and $B$ shows that in that case $AB - BA = 0$ if and only if all their projections, $\{P_\Omega^A\}$ and $\{P_\Omega^B\}$, commute, We take this as our definition in the unbounded case.

**Definition 5.5.1.** Two possibly unbounded in $*\mathbb{R}_c^\#$ self-#-adjoint operators $A$ and $B$ are said to commute if and only if all the projections in their associated projection-valued #-measures commute.

**Remark 5.5.1.** The spectral theorem shows that if $A$ and $B$ commute, then all the bounded in $*\mathbb{R}_c^\#$ #-Borel functions of $A$ and $B$ also commute. In particular, the resolvents $R_\lambda(A)$ and $R_\mu(B)$ commute and the unitary groups $\mathrm{Ext}\text{-exp}(itA)$ and $\mathrm{Ext}\text{-exp}(isA)$ commute.

The converse statement is also true and this shows that the above definition of "commute" is reasonable:

**Theorem 5.5.1.** Let $A$ and $B$ be self-#-adjoint operators on a non-Archimedean Hilbert space $\mathcal{H}$.

Then the following three statements are equivalent:

(a) Spectral projections $P_{(a,b)}^A$ and $P_{(c,d)}^B$, commute.

(b) If $\mathrm{Im} \lambda$ and $\mathrm{Im} \mu$ are nonzero, then $R_\lambda(A)R_\mu(B) - R_\mu(B)R_\lambda(A) = 0$.

(c) For all $s, t \in *\mathbb{R}_c^\#$, $[\mathrm{Ext}\text{-exp}(itA)][\mathrm{Ext}\text{-exp}(isB)] = [\mathrm{Ext}\text{-exp}(isB)][\mathrm{Ext}\text{-exp}(itA)]$.

**Proof** The fact that (a) implies (b) and (c) follows from the functional calculus. The fact that (b) implies (a) easily follows from the formula which expresses the spectral projections of $A$ and $B$ as strong #-limits of the resolvents (generalized Stone's formula) together with the fact that

$$s\text{-#-}\lim_{\epsilon \to 0} [i\epsilon R_{a+i\epsilon}(A)] = P_{\langle a \rangle}^A. \qquad (5.5.1)$$

To prove that (c) implies (a), we use some simple facts about the Fourier transform. Let $f \in S^#(*\mathbb{R}_c^\#)$. Then, by generalized Fubini's theorem,

$$ \begin{aligned} \mathrm{Ext} - \int_{*\tilde{\mathbb{R}}_c^\#} f(t) \langle [\mathrm{Ext}\text{-exp}(itA)]\varphi, \psi \rangle_\# d^\#t &= \mathrm{Ext} - \int_{*\tilde{\mathbb{R}}_c^\#} f(t) \left( \mathrm{Ext} - \int_{*\tilde{\mathbb{R}}_c^\#} ([\mathrm{Ext}\text{-exp}(-it\lambda)]d_\lambda^\# \langle P_\lambda^A \varphi, \psi \rangle_\#) \right) d^\#t \\ &= \sqrt{2\pi\#} \left( \mathrm{Ext} - \int_{*\tilde{\mathbb{R}}_c^\#} \hat{f}(\lambda) d_\lambda^\# \langle P_\lambda^A \varphi, \psi \rangle_\# \right) = \sqrt{2\pi\#} \langle \varphi, \hat{f}(A)\psi \rangle_\# \end{aligned} \qquad (5.5.2) $$

Thus, using (c) and generalized Fubini's theorem again,

$$ \begin{aligned} \langle \varphi, \hat{f}(A) \hat{g}(B) \psi \rangle_\# &= \mathrm{Ext} - \int_{*\tilde{\mathbb{R}}_c^\#} \mathrm{Ext} - \int_{*\tilde{\mathbb{R}}_c^\#} f(t) g(s) \langle \varphi, [\mathrm{Ext}\text{-exp}(-itA)][\mathrm{Ext}\text{-exp}(-isB)] \psi \rangle_\# d^\#s d^\#t \\ &= \langle \varphi, \hat{g}(B) \hat{f}(A) \psi \rangle_\# \end{aligned} \qquad (5.5.3) $$

so, for all $f, g \in S^#(*\mathbb{R}_c^\#)$, $\hat{f}(A)\hat{g}(B) - \hat{g}(B)\hat{f}(A) = 0$.

Since the Fourier transform maps $S^#(*\mathbb{R}_c^\#)$ onto $S^#(*\tilde{\mathbb{R}}_c^\#)$ we conclude that $f(A)g(B) = g(B)f(A)$ for all $f, g \in S^#(*\tilde{\mathbb{R}}_c^\#)$. But, the characteristic function, $\chi_{(a,b)}$ can be expressed as the pointwise #-limit of a hyperinfinite sequence $f_n, n \in *\mathbb{N}$ of uniformly bounded functions in $S^#(*\tilde{\mathbb{R}}_c^\#)$. By the functional calculus,
$$
H_m^\# = \{p \in \mathbb{R}_c^{\#4} p \cdot \tilde{p} = m^2, p_0 > 0\}, \qquad (23)
$$

where $\tilde{p} = (p^0, -p^1, -p^2, -p^3)$. The sets $H_m^\#$, which are called mass hyperboloids, are
invariant under $\sigma \mathcal{L}_+^\uparrow$. Let $j_m$ be the \#-homeomorphism of $H_m^\#$ onto $*\mathbb{R}_c^{\#3}$ (or in the case
$m=0$ onto $*\mathbb{R}_c^{\#3} \setminus \{0\}$) given by $j_m : \langle p_0, p_1, p_2, p_3 \rangle \mapsto \langle p_1, p_2, p_3 \rangle = \mathbf{p}$. Define a
\#-measure $\Omega_m^\#$ on $H_m^\#$ by

$$
\Omega_m^\#(E) = \mathrm{Ext} - \int_{j_m(E)} \frac{d^{\#3} \mathbf{p}}{\sqrt{|\mathbf{p}|^2 + m^2}} \quad (24)
$$

for any measurable set $E \subset H_m^\#$. The measure $\Omega_m^\#(E)$ can easily be seen to be
$\sigma\mathcal{L}_+^\uparrow$-invariant. In fact, up to a constant multiple, $\Omega_m^\#$ is the only $\sigma\mathcal{L}_+^\uparrow$-invariant measure
on $H_m^\#$. Furthermore, every polynomially bounded $\sigma\mathcal{L}_+^\uparrow$-invariant measure on $\bar{V}_+$ is the
sum of a multiple of $\delta$ and an integral of the measures $\Omega_m^\#$. We state this fact as a
theorem.

**Theorem 1.2.** Let $\mu^\#$ be a polynomially bounded \#-measure with support in $\bar{V}_+$. If $\mu^\#$ is $\sigma\mathcal{L}_+^\uparrow$-invariant, there exists a polynomially bounded \#-measure $\rho$ on $[0, \infty^\#)$ and a constant $c$ so that for any $f \in S^\#(*\mathbb{R}_c^{\#4})$

$$
\mathrm{Ext} - \int_{\mathbb{R}_c^{\#4}} f d^\# \mu^\# = cf(0) + \mathrm{Ext} - \int_0^{\infty^\#} d^\# \rho^\#(m) \left( \mathrm{Ext} - \int_{H_m^\#} f d^\# \Omega_m^\# \right). \quad (25)
$$

**Theorem 1.3.**

We can now use the Segal quantization to define the free Hermitian scalar field of
mass $m$. We take $\mathbf{H}^\# = \mathcal{L}_2^\#(H_m^\#, d^\#\Omega_{m,x})$, where $H_m^\#, m > 0$, is the mass hyperboloid in
$*\mathbb{R}_c^{\#4}$ consisting of those $p \in *\mathbb{R}_c^{\#4}$ satisfying $p \cdot \tilde{p} - m^2 = 0$ and $p_0 > 0$, and $d^\#\Omega_m^\#$
is the Lorentz invariant \#-measure.

For each $f \in S^\#(*\mathbb{R}_c^{\#4})$ we define $Ef \in \mathbf{H}^\#$ by $Ef = 2\pi_\# \hat{f} \upharpoonright H_m^\#$ where the Fourier transform

$$
(2\pi_{\#})^{-2} \left( \mathrm{Ext} - \int \left( \mathrm{Exp} - \exp \left[ i \left( p \cdot \tilde{x} \right) \right] \right) f(x) d^{\#4} x \right) \quad (26)
$$

is defined in terms of the Lorentz invariant inner product $p \cdot \tilde{x}$. The reason for the
extra $\sqrt{2\pi_\#}$ in our definition of $E$ and the plus sign in the definition of Fourier transform
is that if $f$ is the distribution $f(x) = g(x)\delta^\#(t)$, then $\sqrt{2\pi_\#} \hat{f}$ is the ordinary
three-dimensional

Fourier transform of $g$. If $\Phi_S^\#(\cdot)$ is the Segal quantization over $\mathcal{L}_2^\#(H_m^\#, d^\#\Omega_{m,x})$, we
define
for each $*\mathbb{R}_c^\#$-valued $f \in S^\#(*\mathbb{R}_c^{\#4})$

$$
\Phi_{m,x}^{\#}(f) = \Phi_{S}^{\#}(Ef). \qquad (27)
$$

For $*\mathbb{C}_c^\#$-valued function $f \in S^\#(*\mathbb{R}_c^{\#4})$ we define

$$
\Phi_{m,x}^{\#}(f) = \Phi_{m,x}^{\#}(\mathrm{Re}f) + i\Phi_{m,x}^{\#}(\mathrm{Im}f) \quad (28)
$$

The mapping $f \mapsto \Phi_m^\#(f)$ is called the free Hermitian scalar field of mass $m$.
On $\mathcal{L}_2^\#(H_m^\#, d^\#\Omega_m)$ we define the following unitary representation of the restricted
Poincare group:

$$
(U_m(a, \Lambda)\psi)(p) = (\mathrm{Exp} - \exp[i(p \cdot \tilde{a})])\psi(\Lambda^{-1}p) \quad (29)
$$

where we are using $\Lambda$ to denote both an element of the abstract restricted Lorentz group
$$s\text{-}\#\text{-}\lim_{n\to\infty} f_n(A) = P_{(a,b)}^A. \qquad (5.5.4)$$

Similarly, we find uniformly bounded $g_n \in S^#(*\tilde{\mathbb{R}}_c^#)$ #-converging pointwise to $\chi_{(c,d)}$ and

$$s\text{-}\#\text{-}\lim_{n\to\infty} g_n(B) = P_{(c,d)}^B. \qquad (5.5.5)$$

Since the $f_n$ and $g_n$ are uniformly bounded in $*\tilde{\mathbb{R}}_c^#$ and

$$f_n(A)g_n(B) = g_n(B)f_n(A) \qquad (5.5.6)$$

for each $n \in *\mathbb{N}$, we conclude that $P_{(a,b)}^A$ and $P_{(c,d)}^B$, commute which proves (a).

**Definition 5.5.2.** Let $A: H^\# \to H^\#$ be bounded in $*\tilde{\mathbb{R}}_c^#$ self-#-adjoint operator. The operator $A$ is **essentially bounded** in $*\tilde{\mathbb{R}}_c^#$ if there is $\text{st}(\|A\|_\#) \in \mathbb{R}$ and $\text{st}(\|A\|_\#) \neq \infty$.

**Remark 5.5.2.** Note that if $A$ is essentially bounded in $*\tilde{\mathbb{R}}_c^#$ operator then for any nearstandard vector $\psi \in H^\#$ vector $A\psi$ again nearstandard, i.e. $\text{st}(\|A\psi\|_\#) \neq \infty$.

**Definition 5.5.3.** Let $A$ and $B$ be self-#-adjoint essentially bounded in $*\tilde{\mathbb{R}}_c^#$ operators on a non-Archimedean Hilbert space $H^\#$. The operators $A$ and $B$ are $\approx$-commute if $\|AB\|_\# \approx \|BA\|_\#$

**Remark 5.5.3.** Note that the operators $A$ and $B$ are $\approx$-commute if for any nearstandard vector $\psi \in H^\# : A\psi \approx B\psi$.

**Theorem 5.5.2.** Let $A$ and $B$ be self-#-adjoint operators on a non-Archimedean Hilbert space $H^\#$ and essentially bounded in $*\tilde{\mathbb{R}}_c^#$. Then the following three statements are equivalent:

(a) Spectral projections $P_{(a,b)}^A$ and $P_{(c,d)}^B$, $\approx$-commute.

(b) If $\text{Im}\,\lambda$ and $\text{Im}\,\mu$ are nonzero, then $R_\lambda(A)R_\mu(B)$ and $R_\mu(B)R_\lambda(A)$ $\approx$-commute.

(c) For all $s, t \in *\tilde{\mathbb{R}}_c^#$, $[\text{Ext}\text{-exp}(itA)][\text{Ext}\text{-exp}(isB)]$ and $[\text{Ext}\text{-exp}(isB)][\text{Ext}\text{-exp}(itA)]$ $\approx$-commute.

**Theorem 5.5.3.** Let $A$ and $B$ be self-#-adjoint operators on a non-Archimedean Hilbert space $H^\#$. Then the following three statements are equivalent:

(a) Spectral projections $P_{(a,b)}^A$ and $P_{(c,d)}^B$, $\approx$-commute.

(b) For all $s, t \in *\tilde{\mathbb{R}}_c^#$, $[\text{Ext}\text{-exp}(itA)][\text{Ext}\text{-exp}(isB)] = [\text{Ext}\text{-exp}(isB)][\text{Ext}\text{-exp}(itA)]$. $\approx$-commute.

## § 5.6. $*\tilde{\mathbb{C}}_c^#$-valued quadratic forms.

One consequence of the generalized Riesz lemma is that there is a one-to-one correspondence between bounded in $*\tilde{\mathbb{R}}_c^#$ quadratic forms and bounded in $*\tilde{\mathbb{R}}_c^#$ operators; that is, any sesquilinear

map $q: H \times H \to *\tilde{\mathbb{C}}_c^#$ which satisfies $|q(\varphi, \psi)_\#| < M\|\varphi\|_\#\|\psi\|_\#$ is of the form $q(\varphi, \psi) = \langle\varphi, A\psi\rangle_\#$ for some bounded operator $A$. As one might expect, the situation is more complicated if one removes the boundedness restriction. It is the relationship between unbounded forms and unbounded operators which we study briefly in this section.

**Definition 5.6.1.** A quadratic form is a map $q: Q(q) \times Q(q) \to *\tilde{\mathbb{C}}_c^#$, where $Q(q)$ is a #-dense linear subset of $H$ called the form domain, such that $q(\cdot, \psi)$ is conjugate linear and $q(\varphi, \cdot)$ is linear for $\varphi, \psi \in Q(q)$. If $q(\varphi, \psi) = \overline{q\langle\psi, \varphi\rangle}_\#$ we say that $q$ is symmetric. If $q(\varphi, \varphi) \ge 0$ for all $\varphi \in Q(q)$, $q$ is called positive, and if $q(\varphi, \varphi) \ge -M\|\varphi\|_\#^2$ for some $M \in *\tilde{\mathbb{R}}_c^#$ we say that $q$ is semibounded in $*\tilde{\mathbb{R}}_c^#$.
and the corresponding element in the standard representation on $*\mathbb{R}_{\text{st}}^4 = \mathbb{R}^4$.

**Remark 1.3.** Recall that a #-conjugation on a Hilbert space $\mathbf{H}^\#$ is an antilinear #-isometry $\mathbf{C}^\#$ so that $\mathbf{C}^{\#2} = \mathbf{I}$.

**Definition 1.2.** Let $\mathbf{H}^\#$ be a $*\mathbb{C}_c^\#$-complex Hilbert space, $\Phi_S^\#(\cdot)$ the associated Segal quantization. Let $\mathbf{C}^\#$ be a #-conjugation on $\mathbf{H}^\#$ and define $\mathbf{H}_{\mathbf{C}^\#} = \{|\mathbf{C}^\#f = f\}$. For each $f \in \mathbf{H}_{\mathbf{C}^\#}^\#$ we define $\varphi^\#(f) = \Phi_S^\#(f)$ and $\pi^\#(f) = \Phi_S^\#(if)$. The map $f \mapsto \varphi^\#(f)$ is called the canonical free field over the doublet $\langle \mathbf{H}^\#, \mathbf{C}^\# \rangle$ and the map $f \mapsto \pi^\#(f)$ is called the canonical conjugate momentum. We often drop the $\langle \mathbf{H}^\#, \mathbf{C}^\# \rangle$ and just write $\mathbf{H}^\#$ if the intended #-conjugation is clear.

**Remark 1.4.** Note that the set of elements of $\mathbf{H}^\#$ for which the maps $f \mapsto \varphi^\#(f)$ and $f \mapsto \pi^\#(f)$ are defined depends on the #-conjugation $\mathbf{C}^\#$.

**Theorem 1.4.** Let $\mathbf{H}^\#$ be a $*\mathbb{C}_c^\#$-complex Hilbert space with #-conjugation $\mathbf{C}^\#$. Let $\varphi^\#(\cdot)$ and $\pi^\#(\cdot)$ be the corresponding canonical fields. Then:

(i) For each $f \in \mathbf{H}_{\mathbf{C}^\#}^\#$, $\varphi^\#(f)$ is essentially self-adjoint on $F_0$.

(ii) $\{\varphi^\#(f)|f \in \mathbf{H}_{\mathbf{C}^\#}^\#\}$ is a commuting family of self-adjoint operators.

(iii) $\Omega_0$ is a #-cyclic vector for the family $\{\varphi^\#(f)|f \in \mathbf{H}_{\mathbf{C}^\#}^\#\}$.

(iv) If $\#\text{lim}_{n \to \infty}^\# f_n = f$ in $\mathbf{H}_{\mathbf{C}^\#}^\#$, then

$$ \#\text{lim}_{n \to \infty}^\# \varphi^\#(f_n)\psi = \varphi^\#(f)\psi \text{ for all } \psi \in F_0 $$

and

$$ \#\text{lim}_{n \to \infty}^\# (\text{Exp}-\text{exp}[i\varphi^\#(f_n)]\psi) = \text{Exp}-\text{exp}[i\varphi^\#(f)]\psi \text{ for all } \psi \in \mathcal{F}_s(\mathbf{H}^\#) $$

(v) Properties (i)-(iv) hold with $\varphi^\#(f)$ replaced by $\pi^\#(f)$.

(vi) If $f, g \in \mathbf{H}_{\mathbf{C}^\#}^\#$, then

$$ \varphi^\#(f)\pi^\#(g)\psi - \pi^\#(g)\varphi^\#(f)\psi = i(f,g)\psi \quad (30) $$

for all $\psi \in F_0$ and

$$ (\text{Exp}-\text{exp}[i\varphi^\#(f)]) (\text{Exp}-\text{exp}[i\pi^\#(g)]) = (\text{Exp}-\text{exp}[i(f,g)]) (\text{Exp}-\text{exp}[i\pi^\#(g)]) (\text{Exp}-\text{exp}[i\varphi^\#(f)]). \quad (31) $$

**Proof.** (i) and (iv) follow immediately from the corresponding properties of $\Phi_S^\#(\cdot)$ proven in Theorem 1.1. To see that $\{\varphi^\#(f)|f \in \mathbf{H}_{\mathbf{C}^\#}^\#\}$ is a commuting family, notice that (12) implies

$$ (\text{Exp}-\text{exp}[it\varphi^\#(f)]) (\text{Exp}-\text{exp}[is\varphi^\#(g)]) = (\text{Exp}-\text{exp}[-its\text{Im}(f,g)]) (\text{Exp}-\text{exp}[is\varphi^\#(g)]) (\text{Exp}-\text{exp}[it\varphi^\#(f)]) \quad (32) $$

where we have used the fact that $\varphi^\#(\cdot)$ is real linear. If $f, g \in \mathbf{H}_{\mathbf{C}^\#}^\#$, then it follows from polarization that $(f, g) = (C^\#f, C^\#g) = (g, f)$, so $\text{Im}(f, g) = 0$. Thus

$$ (\text{Exp}-\text{exp}[it\varphi^\#(f)]) (\text{Exp}-\text{exp}[is\varphi^\#(g)]) = (\text{Exp}-\text{exp}[is\varphi^\#(g)]) (\text{Exp}-\text{exp}[it\varphi^\#(f)]) \quad (33) $$

for $s$ and $t$. Therefore, by Theorem VIII. 13, $\varphi^\#(g)$ and $\varphi^\#(f)$ commute.

The proof of (b) is similar to the proof of (a). (X.70) and (X.71) follow immediately from (X.64), (X.65), and the fact that if $f, g \in \mathbf{H}_{m,x}^\#$, then $\text{Im}(f, ig) = \text{Re}(f, g) = (f, g)$.

**Definition 1.3.** We write $f \in \mathcal{L}_2^\#(H_m^\#, d^\#\Omega_{m,x}^\#)$ as $f(p_0, \mathbf{p})$ and define now the #-conjugation by $(\mathbf{C}^\#f)(p_0, \mathbf{p}) = \overline{f(p_0, -\mathbf{p})}$.
Notice that if $q$ is semibounded, then it is automatically symmetric if $H$ is complex.

**Example 5.6.1.** Let $H = \mathcal{L}_2^{\#}(\ast\mathbb{R}_c^{\#})$ and $Q(q) = C_0^{\infty}(\ast\mathbb{R}_c^{\#})$ with $q(f,g) = f(0)g(0)$. Then $q$ is a positive quadratic form. Since $q(f,g) = \delta^{\#}(fg)$ one could formally write
$$q(fg) = (f, Ag)$$

where $A: g \mapsto \delta^{\#}(x)g(x)$. Since multiplication by $\delta(x)$ is not an operator, $q$ is an example

of a quadratic form not likely to be associated with an operator.

**Example 5.6.2** Let $A$ be a self-#-adjoint operator on $H^{\#}$. Let us pass to a spectral representation of $A$, so that $A$ is multiplication by $x$ on $\otimes_{n=1}^{N} \mathcal{L}_2^{\#}(\ast\tilde{\mathbb{R}}_c^{\#}, \mu_n^{\#})$. Let

$$Q(q) = \{ (\psi_n)_{n=1}^N | \mathrm{Ext} - \sum_{n=1}^N \mathrm{Ext} - \int_{\ast\tilde{\mathbb{R}}_c^{\#}} |x| \|\psi_n(x)\|^2 d^{\#} \mu_n^{\#} \} < *\infty \quad (5.6.1)$$

and for $\varphi, \psi \in Q(q)$ define

$$q(\varphi, \psi) = \sum_{n=1}^{N} \left( \mathrm{Ext} - \int_{\ast\tilde{\mathbb{R}}_c^{\#}} x \overline{\varphi_n(x)} \psi_n(x) d^{\#} \mu_n^{\#} \right). \quad (5.6.2)$$

We call $q$ the quadratic form associated with $A$ and write $Q(q) = Q(A)$; $Q(A)$ is called the form domain of the operator $A$. For $\psi, \varphi \in Q(A)$, we will write $q(\varphi, \psi) = \langle \varphi, A\psi \rangle_{\#}$ although $A$ does not make sense on all $\psi \in Q(A)$, then $Q(A)$ is in some sense the largest domain on which $q$ can be defined.

To investigate the deep connection between self-#-adjointness and semi-bounded in $\ast\mathbb{R}_c^{\#}$ quadratic forms we need to extend the notion of “#-closed” from operators to forms. An operator $A$ is #-closed if and only if its graph is #-closed which is the same as saying that $D(A)$ is complete under the #-norm $\|\psi\|_A = \|A\psi\|_\# + \|\psi\|_\#$.

Analogously we define:

**Definition 5.6.2.** Let $q$ be a semibounded in $\ast\mathbb{R}_c^{\#}$ quadratic form, $q(\psi, \psi) \ge -M\|\psi\|_\#^2$ is called #-closed if $Q(q)$ is complete under the #-norm

$$\|\psi\|_{\#+1} = \sqrt{q(\psi, \psi) + (M+1)\|\psi\|_{\#}^2}. \quad (5.6.3)$$

If $q$ is #-closed and $D \subset Q(q)$ is #-dense in $Q(q)$ in the $\|\psi\|_{\#+1}$ #-norm, then $D$ is called a form #-core for $q$.

Notice that $\|\psi\|_{\#+1}$ comes from the inner product

$$\langle \psi, \varphi \rangle_{\#+1} = q(\psi, \varphi) + (M+1) \langle \psi, \varphi \rangle_{\#}. \quad (5.6.4)$$

It is not hard to see that $q$ is #-closed if and only if whenever

$\varphi_n \in Q(q)$ $\varphi_n \xrightarrow{H^{\#}} \varphi$ and $q(\varphi_n - \varphi_m, \varphi_n - \varphi_m) \xrightarrow{\#} 0$, as $n, m \to *\infty$, then $\varphi \in Q(q)$ and $q(\varphi_n - \varphi, \varphi_n - \varphi) \xrightarrow{\#} 0$. This criterion and the dominated #-convergence theorem show that the form $q$ associated with a semibounded self-#-adjoint operator (Example 5.6.2) is #-closed. Furthermore, any operator #-core for $A$ is a form #-core for $q$.

Now, let $q(f,g) = f(0)g(0)$ as in Example 5.6.1 and $\varphi_n \in C_0^{\infty}(\ast\tilde{\mathbb{R}}_c^{\#})$. Then $\varphi_n \to 0$, and $q(\varphi_n - \varphi_m, \varphi_n - \varphi_m) \to \# 0$, but $q(\varphi_n, \varphi_n) \to \# 1 \neq q(0,0)$ which proves that $q$ has no #-closed extensions. Therefore, even though $q$ is positive (and therefore symmetric) there is no semibounded self-#-adjoint operator $A$ so that $q(f,g) = \langle f, Ag \rangle_\#$ for all $f,g \in C_0^{\infty}(\ast\tilde{\mathbb{R}}_c^{\#})$.

The deep fact about semibounded quadratic forms is that unlike the case for operators,

they cannot be #-closed and symmetric, yet not self-#-adjoint.
**Remark 1.4.** Note that $\mathbf{C}^{\#}$ is well-defined on $\mathcal{L}_{2}^{\#}(H_{m}^{\#}, d\Omega_{m,x}^{\#})$ since $\langle p_{0}, \mathbf{p} \rangle \in H_{m}^{\#}$ if and only if $\langle p_{0}, -\mathbf{p} \rangle \in H_{m}^{\#}$. $\mathbf{C}^{\#}$ is clearly a $\#$-conjugation.

**Definition 1.4.** We denote the canonical fields corresponding to $\mathbf{C}^{\#}$ by $\varphi^{\#}(\cdot)$ and $\pi^{\#}(\cdot)$ and define $\varphi_{m}^{\#}(f) = \varphi^{\#}(Ef)$ and $\pi_{m}^{\#}(f) = \pi^{\#}(\mu Ef)$, $\mu = \sqrt{\mathbf{p}^{2} + m^{2}}$ for $^{*}\mathbb{R}_{c}^{\#}$-valued $f \in \mathcal{L}(^{*}\mathbb{R}_{c}^{\#4})$, extending to all of $\mathcal{L}(^{*}\mathbb{R}_{c}^{\#4})$ by linearity. In terms of $a^{-}(f)$,

$$ 
\begin{aligned}
\varphi_{m}^{\#}(f) &= \{(a^{-}(Ef))^{*} + a^{-}(\mathbf{C}^{\#}Ef)\}/\sqrt{2}, \\
\pi_{m}^{\#}(f) &= i\{(a^{-}(Ef))^{*} + a^{-}(\mathbf{C}^{\#}\mu Ef)\}/\sqrt{2}.
\end{aligned} 
\quad (34) $$

**Remark 1.5.** Note that the $a$'s in these last formulas differ from those most often used in discussing the free field and that the correct space-time free field is $\mathbf{\Phi}_{m}^{\#}$ and not $\varphi_{m}^{\#}$ as we will discuss below, $\varphi_{m}^{\#}$ and $\pi_{m}^{\#}$ are useful for discussing the time-zero field. The maps $f \mapsto \varphi_{m}^{\#}(f)$ and $f \mapsto \pi_{m}^{\#}(f)$ are complex linear and $\varphi_{m}^{\#}(f), \pi_{m}^{\#}(f)$ are self-adjoint if and only if $Ef \in \mathbf{H}_{c}^{\#}$.

Because of the projection $E$ we can extend the class of functions on which $\varphi_{m}^{\#}(\cdot)$ and $\pi_{m}^{\#}(\cdot)$ are defined to include distributions of the form $\delta(t - t_{0})g(x_{1}, x_{2}, x_{3})$ where $g \in ^{*}\mathbb{R}_{c}^{\#3}$. In particular, if $t_{0} = 0$, $g$ is $^{*}\mathbb{R}_{c}^{\#}$-valued, and $\mathrm{Ext}\_\hat{g}$ is the usual Fourier transform on $^{*}\mathbb{R}_{c}^{\#3}$, then

$$ (\mathbf{C}^{\#}E\hat{\delta g})(p_{0}, -\mathbf{p}) = (2\pi_{\#})^{-1/2}\overline{\hat{g}(-\mathbf{p})} = (2\pi_{\#})^{-1/2}\hat{g}(-\mathbf{p}) = E\hat{\delta g}. \quad (35) $$

Thus $E(\delta g)$ and $\mu E(\delta g)$ are in $\mathbf{H}_{c}^{\#\#}$. Therefore $\varphi_{m}^{\#}(\delta g)$ and $\pi_{m}^{\#}(\delta g)$ are self-adjoint if $g \in \mathcal{L}(^{*}\mathbb{R}_{c}^{\#3})$ is real. For obvious reasons, the maps $g \mapsto \varphi_{m}^{\#}(\delta g)$, $g \mapsto \pi_{m}^{\#}(\delta g)$ are called the time-zero fields. From now on we will only use test functions of the form $\delta g$ in $\varphi_{m}^{\#}(\cdot)$ and $\pi_{m}^{\#}(\cdot)$ and write $\varphi_{m}^{\#}(g)$ and $\pi_{m}^{\#}(g)$ if $g \in S^{*}\mathbb{R}_{c}^{\#3}$ instead of $\varphi_{m}^{\#}(\delta g)$ and $\pi_{m}^{\#}(\delta g)$.

If $f$ and $g$ are $^{*}\mathbb{R}_{c}^{\#}$-valued functions in $\mathcal{L}(^{*}\mathbb{R}_{c}^{\#3})$, then (X.70) implies that for $\psi \in F_{0}$,

$$ [\varphi_{m}^{\#}(f), \pi_{m}^{\#}(g)]\psi = i\left(\mathrm{Ext} - \int_{H_{m}} \overline{\hat{f}(p)}\hat{g}(p)\mu(p)\psi d\Omega_{m,x}^{\#}\right). \quad (36) $$

For convenience and also so that our notation coincides with the standard terminology,

we now transfer the fields we have constructed from the Fock space built up from $\mathcal{L}_{2}^{\#}(H_{m}^{\#}, d\Omega_{m,x}^{\#})$ to the Fock space built up from $\mathcal{L}_{2}^{*}(^{*}\mathbb{R}_{c}^{\#3})$. For notational simplicity, we define for $f \in \mathcal{L}_{2}^{\#}(H_{m}^{\#}, d\Omega_{m,x}^{\#})$

$$ a^{\dagger}(f) = (a^{-}(f))^{*}, a(f) = a^{-}(\mathbf{C}^{\#}f). \quad (37) $$

First notice that each function $f(p) \in \mathcal{L}_{2}^{\#}(H_{m}^{\#}, d\Omega_{m,x}^{\#})$ is in a natural way a function $f(\mathbf{p}) = f(\mu(\mathbf{p}), \mathbf{p})$ on $^{*}\mathbb{R}_{c}^{\#3}$. For each $f \in \mathcal{L}_{2}^{\#}(H_{m}^{\#}, d\Omega_{m,x}^{\#})$, we define

$$ (Jf)(\mathbf{p}) = f(\mu(\mathbf{p}), \mathbf{p})/\sqrt{\mu(\mathbf{p})}. \quad (38) $$

$J$ is a unitary map of $\mathcal{L}_{2}^{\#}(H_{m}^{\#}, d\Omega_{m,x}^{\#})$ onto $\mathcal{L}_{2}^{*}(^{*}\mathbb{R}_{c}^{\#3})$, so $\Gamma^{\#}(J)$ is a unitary map of $\mathcal{F}_{s}(\mathcal{L}_{2}^{\#}(H_{m}^{\#}, d\Omega_{m,x}^{\#}))$ onto $\mathcal{F}_{s}(\mathcal{L}_{2}^{*}(^{*}\mathbb{R}_{c}^{\#3}))$. The annihilation and creation operators on $\mathcal{F}_{s}(\mathcal{L}_{2}^{\#}(^{*}\mathbb{R}_{c}^{\#3}))$, $\bar{a}(\cdot)$, $\bar{a}^{\dagger}(\cdot)$, are related to $a(\cdot)$ and $a^{\dagger}(\cdot)$ by the formulas
**Theorem 5.6.2.** If $q$ is a \#-closed semibounded in $*\mathbb{R}_c^\#$ quadratic form, then $q$ is the quadratic form of a unique self-\#-adjoint operator.

**Proof** We may assume without loss of generality that $q$ is positive. Then, since $q$ is \#-closed and symmetric, $Q(q)$ is a Hilbert space, which we denote by $H_{+1}^\#$, under the inner product $\langle\phi, \psi\rangle_{\#+1} = q(\phi, \psi) + \langle\phi, \psi\rangle_\#$. We denote by $H_{-1}^\#$ the space of bounded in $*\mathbb{R}_c^\#$ conjugate linear functionals on $H_{+1}^\#$. Let $j$, given by $\psi \stackrel{j}{\mapsto} \langle\cdot, \psi\rangle_\#$ be the linear imbedding of $H^\#$ into $H_{-1}^\#$ is bounded in $*\mathbb{R}_c^\#$ because

$$|[j(\psi)(\phi)]| \le \|\phi\|_\# \|\psi\|_\# \le \|\phi\|_\# \|\psi\|_\# \le \|\phi\|_{\#+1} \|\psi\|_\#. \quad (5.6.5)$$

Since the identity map $i$ embeds $H_{+1}^\#$ in $H^\#$ we have a “scale of spaces”

$$H_{+1}^{\#} \xrightarrow{i} H^{\#} \xrightarrow{j} H_{-1}^{\#}. \qquad (5.6.6)$$

We now exploit the generalized Riesz lemma. Given $\Phi \in H_{-1}^\#$, let $\hat{B}\Phi$ be the element of $H_{-1}^\#$ which acts by $[\hat{B}\Phi](\phi) = q(\phi, \Phi) + (\phi, \Phi)_\#$. By the generalized Riesz lemma, $\hat{B}$ is an isometric isomorphism of $H_{+1}^\#$ onto $H_{-1}^\#$. Let $D(B) = \{\psi \in H_{+1}^\# | \hat{B}\psi \in \text{Ran}(j)\}$.

Define now $B$ on $D(B)$ by $B = j^{-1}\hat{B}$. Notice that

$$H^{\#} \supset H_{+1}^{\#} \xrightarrow{B} H_{-1}^{\#} \xleftarrow{j} H^{\#}. \qquad (5.6.7)$$

First, we prove that the range of $j$ is \#-dense in $H_{-1}^\#$. If it were not, there would be a $\lambda \in H_{-1}^{\#*}$ so that $\lambda \ne 0$, and $\lambda[j(\psi)] = 0$ for each $\psi \in H^\#$. By the generalized Riesz Lemma, there is a $\phi_\lambda \ne 0$ in $H_{+1}^\#$ so that $0 = \lambda[j(\psi)] = [j(\psi)](\phi_\lambda) = \langle\phi_\lambda, \psi\rangle_\#$ for all $\psi \in H^\#$. Since $\phi_\lambda \ne 0$, this is impossible. Therefore $\text{Ran}(j)$ is \#-dense in $H_{-1}^\#$. Since $B$ is an isometric isomorphism we conclude that $D(B)$ is $\|\cdot\|_{\#+1}$ \#-dense in $H_{+1}^\#$. Further, since $\|\cdot\|_\# \le \|\cdot\|_{\#+1}$ and $H_{-1}^\#$ is \#-norm \#-dense in $H^\#$, $D(B)$ is \#-norm \#-dense in $H^\#$. Suppose $\phi, \psi \in D(B)$. Then one obtains that

$$\langle\phi, B\psi\rangle_\# = q(\phi, \psi) + \langle\phi, \psi\rangle_\# = \overline{q(\psi, \phi) + \langle\psi, \phi\rangle_\#} = \overline{\langle\psi, B\phi\rangle_\#} = \langle B\phi, \psi\rangle_\#. \quad (5.6.8)$$

Thus, $B$ is a \#-densely defined symmetric operator.

We will prove now that $B$ is self-\#-adjoint. Let $C = (\hat{B})^{-1}$. $C$ takes $H^\#$ into $H^\#$ and is an everywhere defined symmetric operator. By the generalized Hellinger-Toeplitz theorem, $C$ is a bounded in $*\mathbb{R}_c^\#$ self-\#-adjoint operator. Moreover, $C$ is injective. A simple application of the spectral theorem in multiplication operator form shows that $C^{-1} : \text{Ran}(C) \to H^\#$ is a self-\#-adjoint operator. But $C^{-1} = B$.

We now define $A = B - I$. Then $A$ is also self-\#-adjoint on $D(A) = D(B)$ and for $\phi, \psi \in D(A)$, $\langle\phi, A\psi\rangle_\# = q(\phi, \psi)$. Since $D(A)$ is $\|\cdot\|_{\#+1}$ \#-dense in $H_{+1}^\#$ is the quadratic form associated to $A$. Uniqueness is obvious.

Thus, there is a principal distinction between semi-bounded in $*\mathbb{R}_c^\#$ symmetric operators and semi-bounded in $*\tilde{\mathbb{R}}_c^\#$ quadratic forms. For symmetric operators, there is never any problem finding \#-closed extensions.

**Reark 5.6.1.** Note that: (1) If $A$ and $B$ are self-\#-adjoint operators and $D(A) \subset D(B)$ with $B \upharpoonright D(A) = A$ then $A = B$. But it can happen that $a$ and $b$ are \#-closed semibounded in $*\mathbb{R}_c^\#$ quadratic forms and $b \upharpoonright Q(a)xQ(a) = a$ without having $a = b$. (2) Let $A$ be a symmetric operator that is semibounded in $*\mathbb{R}_c^\#$. Let $q$ be the quadratic form $q(\phi, \psi) = \langle\phi, A\psi\rangle_\#$ with $Q(a) = D(A)$. Suppose that $q$ has a \#-closure, that is, a smallest \#-closed form which extends it. Then the self-\#-adjoint operator $A$ which
$$
\begin{align}
\tilde{a} \left( \frac{f(\mathbf{p})}{\sqrt{\mu(\mathbf{p})}} \right) &= \Gamma^{\#}(J) a(f) \Gamma^{\#}(J)^{-1} \tag{39} \\
\tilde{a}^{\dagger} \left( \frac{f(\mathbf{p})}{\sqrt{\mu(\mathbf{p})}} \right) &= \Gamma^{\#}(J) a^{\dagger}(f) \Gamma^{\#}(J)^{-1}
\end{align}
$$

We use the unitary map $\Gamma^{\#}(J)$ to carry the Wightman fields over to $\mathcal{F}_s(\mathcal{L}_2^{\#}(\mathbb{R}_c^{\#3}))$ by defining:(i) for $\mathbb{R}_{c,\text{fin}}^{\#}$-valued $f \in \mathcal{L}_{\text{fin}}^{\#}(\mathbb{R}_c^{\#4})$

$$
\begin{equation}
\begin{split}
\tilde{\Phi}_{m,x}(f) ={}& \Gamma^{\#}(J) \Phi_{m,x}(f) \Gamma^{\#}(J)^{-1} = \\
& \frac{1}{\sqrt{2}} \left\{ \tilde{a} \left( \tilde{\mathbf{C}}^{\#} \frac{Ef}{\sqrt{\mu}} \right) + \tilde{a}^{\dagger} \left( \frac{Ef}{\sqrt{\mu}} \right) \right\}
\end{split}
\tag{40}
\end{equation}
$$

(ii) for $\mathbb{R}_{c,\text{fin}}^{\#}$-valued $f \in \mathcal{L}_{\text{fin}}^{\#}(\mathbb{R}_c^{\#3})$

$$
\begin{equation}
\begin{split}
\tilde{\varphi}_{m,x}(f) ={}& \Gamma^{\#}(J) \varphi_{m,x}(f) \Gamma^{\#}(J)^{-1} = \\
& \frac{1}{\sqrt{2}} \left\{ \tilde{a} \left( \tilde{\mathbf{C}}^{\#} \frac{E(f\delta)}{\sqrt{\mu}} \right) + \tilde{a}^{\dagger} \left( \frac{E(f\delta)}{\sqrt{\mu}} \right) \right\}
\end{split}
\tag{41}
\end{equation}
$$

where $\tilde{\mathbf{C}}^{\#} = J\mathbf{C}^{\#}J^{-1}$ acts by $(\tilde{\mathbf{C}}^{\#}g)(\mathbf{p}) = \overline{g(-\mathbf{p})}$. Having established this correspondence,

we now drop the ~ and the bold face letters; from now on we will only deal with the
fields

on $\mathcal{F}_s(\mathcal{L}_2^{\#}(\mathbb{R}_c^{\#3}))$ and three-dimensional momenta. Further, we recall that the
restriction of

the four-dimensional Fourier transform that we have been using in this section to
functions of the form $\delta(x_0)g(x_1, x_2, x_3)$ the usual three-dimensional Fourier transform.

Notice that

$$
\tilde{f} = \mathrm{Ext-}\tilde{h}, h = (C^{\#}\hat{f}) \quad (42)
$$

so $\mathbf{C}^{\#} \hat{f} = \hat{f}$ if and only if $f$ is $\mathbb{R}^{\#}$-valued.

For $f$ and $g$ $\mathbb{R}^\#$-valued, (36) becomes

$$
[\varphi_m^{\#}(f), \pi_m^{\#}(g)] \approx i(\mathrm{Ext} - \int f(x)g(x))d^{\#3}x. \quad (43)
$$

(43) is the space form of the canonical commutation relations (CCR).

In the Appendix to this section we prove that for each $m > 0$, this representation of the CCR is irreducible and for different $m$, the representations are inequivalent. Thus, the time-zero fields in the free scalar field theories give rise to different representation of the

CCR.

As a final topic before turning to interacting fields we will show how the structures
developed above are related to the “fields” and “annihilation and creation
operators” introduced in physics texts. We let now

$$
D_{S_{\text{fin}}^{\#}} = \{ \psi | \psi \in F_0, \psi^{(n)} \in S_{\text{fin}}^{\#} (*\mathbb{R}_c^{\#3n}), n \in \mathbb{N} \} \quad (44)
$$

and for each $p \in \mathbb{R}_c^{\#3}$ we define an operator $a(p)$ on $\mathcal{F}_s(\mathcal{L}_2^{\#}(\mathbb{R}_c^{\#3}))$ with domain $D_{S_{\text{fin}}^{\#}}$ by
corresponds to $\hat{q}$ (by Theorem 5.6.2) may be bigger than the operator #-closure of A.
(3) While a general quadratic form may have no #-closed extensions, forms that come
directly from semibounded in $^*\tilde{\mathbb{R}}_c^#$ operators always have #-closures and thus
semibounded in $^*\mathbb{R}_c^#$ operators always have self-#-adjoint extensions.

§ 5.7. #-Convergence of unbounded in $^*\tilde{\mathbb{R}}_c^#$ operators

One of the main difficulties with unbounded in $^*\mathbb{R}_c^#$ operators is that they are only
#-densely defined. This difficulty is especially troublesome when one wants to find a
notion of #-convergence for a hyper infinite sequence $A_n \to_# A, n \in \mathbb{N}$ of unbounded
in $^*\mathbb{R}_c^#$ operators since the domains of the operators $A_n$ may have no vector in
common. For example, if $A_n = (1-n^{-1})x$ on $L_2(^*\tilde{\mathbb{R}}_c^#)$, it is clear that in some sense
$A_n \to_# A = x$; yet we could have been given domains $D(A_n)$ and $D(A)$ of essential
self-#-adjointness for these operators which have no nonzero vector in common. Of
course, in this simple case the #-closures of $A_n$ and $A$ all have the same domain, but
in general this will not be true, and in any case, one is often forced to deal with
domains of essential self-adjointness since closures of operators are sometimes
difficult to compute. It is very natural to say that self-#-adjoint operators are "close" if
certain bounded in $^*\tilde{\mathbb{R}}_c^#$ functions of them are "close." Most of this section is devoted
to this approach. However, we also introduce graph #-limits, a topic which will be
explored further.

**Definition 5.7.1.** Let $(A_n)_{n \in \mathbb{N}}$ and $A$ be self-#-adjoint operators. Then $A_n$ is said to #-converge to $A$ in the #-norm resolvent sense (or #-norm generalized sense) if $R_\lambda(A_n) \to_# R_\lambda(A)$ in #-norm for all $\lambda$ with $\mathrm{Im}\,\lambda \neq 0$. $A_n$ is said to #-converge to $A$ in the strong resolvent sense (or strong generalized sense) if $R_\lambda(A_n) \to_# R_\lambda(A)$ strongly for all $\lambda$ with $\mathrm{Im}\,\lambda \neq 0$.

We have not introduced the notion of weak resolvent #-convergence since weak
resolvent #-convergence implies strong resolvent #-convergence. The following
theorem shows that #-norm resolvent #-convergence is the right generalization of
#-norm convergence for bounded in $^*\mathbb{R}_c^#$ self-#-adjoint operators. A similar result
holds for strong resolvent #-convergence, but the analogue for weak #-convergence
is not true.

**Theorem 5.7.1.** Let $(A_n)_{n \in \mathbb{N}}^{*\infty}$ and $A$ be a family of uniformly bounded in $^*\tilde{\mathbb{R}}_c^#$ self-#-adjoint operators. Then $A_n \to_# A$ as $n \to *\infty$ in the #-norm resolvent sense if and only if $A_n \to_# A$ as $n \to *\infty$ in #-norm.

**Proof.** Let $A_n \to_# A$ as $n \to *\infty$ in #-norm. Then if $\mathrm{Im}\,\lambda \neq 0$, $(A_n - A)(A - \lambda)^{-1} \to_# 0$ in #-norm. Thus, using the equality $(A_n - \lambda)^{-1} = (A - \lambda)^{-1} [I + (A_n - A)(A - \lambda)^{-1}]^{-1}$ we obtain that $(A_n - \lambda)^{-1} \to_# (A - \lambda)^{-1}$ in #-norm as $n \to *\infty$.

Conversely, suppose $A_n \to_# A$ as $n \to *\infty$ in the #-norm resolvent sense. Then, since $A_n - A = (A_n - i)(A_n - i)^{-1}[(A - i)^{-1} - (A_n - i)^{-1}](A - i)$, we conclude that $\|A_n - A\|_* \le (\sup_n \|A_n\|_* + 1)\|(A - i)^{-1} - (A_n - i)^{-1}\|_* (\|A\|_* + 1) \to_# 0$ as $n \to *\infty$. The following theorem shows that to prove generalized convergence one need only show #-convergence of the resolvents at one point off the hyperreal axis $^*\mathbb{R}_c^#$.

**Theorem 5.7.2.** Let $(A_n)_{n=1}^\infty$ and $A$ be self-#-adjoint operators, and let $\lambda_0 \in *C_c^#$.

(a) If $\mathrm{Im}\,\lambda_0 \neq 0$ and $\|R_{\lambda_0}(A_n) - R_{\lambda_0}(A)\|_* \to_* 0$, then $A_n \to_* A$ as $n \to *\infty$ in the #-norm resolvent sense.

(b) If $\mathrm{Im}\,\lambda_0 \neq 0$ and if $R_{\lambda_0}(A_n)\varphi - R_{\lambda_0}(A)\varphi \to_* 0$, for all $\varphi \in H^*$ then $A_n \to_* A$ as
$$ (a(p)\psi)^{(n)}(k_1, \dots, k_n) = \sqrt{n+1} \psi^{(n+1)}(p, k_1, \dots, k_n). \quad (45) $$

The adjoint of the operator $a(p)$ is not a $\#$-densely defined operator since it is given formally by

$$ (a^{\dagger}(p)\psi)^{(n)}(k_1, \dots, k_n) = \frac{1}{\sqrt{n}} \sum_{i=1}^{n} \delta(p - k_i)\psi^{(n+1)}(p, k_1, \dots, k_{i-1}, k_{i+1}, \dots, k_n). \quad (46) $$

However, $a^{\dagger}(p)$ is a well-defined quadratic form on $D_{\mathcal{L}_{\text{fin}}^{\#}} \times D_{\mathcal{L}_{\text{fin}}^{\#}}$. For example, if $\psi_1 = \{0, \psi^{(1)}, 0, \dots\}$, and $\psi_2 = \{0, 0, \psi^{(2)}, 0, \dots\}$, then

$$ (\psi_2, a^{\dagger}(p)\psi_1) = \frac{1}{\sqrt{2}} \{ \text{Ext-} \int [\overline{\psi^{(2)}(k_1, p)}\psi^{(1)}(k_1) + \overline{\psi^{(2)}(p, k_1)}\psi^{(1)}(k_1)] d^{\#}k_1 \}. \quad (47) $$

**Remark 1.1.** Note that the formulas

$$ a(g) = \text{Ext-} \int_{\mathbb{R}_c^{\#3}} a(p)g(-p)d^{\#}p \quad (48) $$

and

$$ a^{\dagger}(g) = \text{Ext-} \int_{\mathbb{R}_c^{\#3}} a^{\dagger}(p)g(p)d^{\#3}p \quad (49) $$

hold for all $g \in S_{\text{fin}}^{\#}(*\mathbb{R}_c^{\#3})$ if the equalities are understood in the sense of quadratic forms. That is, (48) means that for $\psi_1, \psi_2 \in D_{S_{\text{fin}}^{\#}}$ we have

$$ (\psi_1, a(g)\psi_2) = \text{Ext-} \int_{\mathbb{R}_c^{\#3}} (\psi_1, a(p)\psi_2)g(-p)d^{\#3}p \quad (50) $$

and similarly for (X.76b).

Since $a(p) : D_{\mathcal{L}_{\text{fin}}^{\#}} \to D_{\mathcal{L}_{\text{fin}}^{\#}}$ the powers of $a(p)$ are well-defined operators on $D_{\mathcal{L}_{\text{fin}}^{\#}}$. As before we can write down a formal expression for $(a^{\dagger}(p))^{n}$, but it does not make sense as operator, only as $*\mathbb{C}_{c}^{\#}$-valued quadratic form on $D_{\mathcal{L}_{\text{fin}}^{\#}} \times D_{\mathcal{L}_{\text{fin}}^{\#}}$.

Notice that

$$ (\psi_1, (a^{\dagger}(p))^{n}\psi_2) = ((a(p))^{n}\psi_1, \psi_2) \quad (51) $$

so for each $n$, $(a^{\dagger}(p))^{n}$ and $(a(p))^{n}$ are formally adjoints in the sense of $*\mathbb{C}_{c}^{\#}$-valued quadratic forms. We could of course have defined the quadratic form $(a^{\dagger}(p))^{n}$ by (50) and then calculated that it arises by taking the $n$-th power of the formal object given by (45). Since $a(p_1) : D_{\mathcal{L}_{\text{fin}}^{\#}} \to D_{\mathcal{L}_{\text{fin}}^{\#}}$, $(\psi_1, a^{\dagger}(p_2)a(p_1)\psi_2)$ is a well-defined $*\mathbb{C}_{c}^{\#}$-valued quadratic form for all $\langle p_1, p_2 \rangle \in *\mathbb{R}_c^{\#3} \times *\mathbb{R}_c^{\#3}$. Notice, however, that $(\psi_1, a(p_1)a^{\dagger}(p_2)\psi_2)$ does not make sense since $a^{\dagger}(p_2)$ is only a quadratic form. In general any product $\prod_{i=1}^{N_1} a(f_i)$ is a well-defined operator from $D_{\mathcal{L}_{\text{fin}}^{\#}}$ to $D_{\mathcal{L}_{\text{fin}}^{\#}}$ and $\prod_{i=1}^{N_1} a^{\dagger}(f_i)$ is a well-defined quadratic form on $D_{\mathcal{L}_{\text{fin}}^{\#}} \times D_{\mathcal{L}_{\text{fin}}^{\#}}$. Thus

$$ \left( \psi_1, \left( \prod_{i=N_1+1}^{N_2} a^{\dagger}(p_i) \right) \left( \prod_{i=1}^{N_1} a^{\dagger}(-p_i) \right) \psi_2 \right) \quad (52) $$

is also well-defined $*\mathbb{C}_{c}^{\#}$-valued quadratic form on $D_{\mathcal{L}_{\text{fin}}^{\#}} \times D_{\mathcal{L}_{\text{fin}}^{\#}}$. One can check directly that if $f \in \mathcal{L}_{\text{fin}}^{\#}(*\mathbb{R}_c^{\#3})$ then as $*\mathbb{C}_{c}^{\#}$-valued quadratic forms
$n \to *\infty$ in the strong resolvent sense.

**Proof** (a) Both $R_\lambda(A)$ and $R_\lambda(A_n)$ are analytic in the half-plane of $*\tilde{\mathbb{C}}_c^\#$ containing $\lambda_0$ and have hyper infinite power series around $\lambda_0$,

$$R_\lambda(A) = \mathrm{Ext} - \sum_{m=0}^{*\infty} (\lambda_0 - \lambda)^m [R_{\lambda_0}(A)]^{m+1}, \qquad (5.7.1)$$

$$R_\lambda(A_n) = \mathrm{Ext} - \sum_{m=0}^{*\infty} (\lambda_0 - \lambda)^m [R_{\lambda_0}(A_n)]^{m+1}$$

which #-converge in #-norm in the circle $|\lambda - \lambda_0| < |\mathrm{Im}\,\lambda_0|^{-1}$. Since $R_{\lambda_0}(A_n) \to_* R_{\lambda_0}(A)$ in #-norm, $R_\lambda(A_n) \to_* R_\lambda(A)$ in #-norm for A in this circle. Therefore, by repeating this process, we get #-convergence for all A in the half-plane of $*\tilde{\mathbb{C}}_c^\#$ containing $\lambda_0$. Furthermore, since

$$\|R_{\bar{\lambda}_0}(A_n) - R_{\bar{\lambda}_0}(A)\|_{\#} = \|(R_{\lambda_0}(A_n) - R_{\lambda_0}(A))^*\|_{\#} = \\ \|R_{\lambda_0}(A_n) - R_{\lambda_0}(A)\|_{\#} \to_* 0 \text{ as } n \to *\infty \qquad (5.7.2)$$

the same argument shows that the resolvents converge in #-norm in the half-plane of $*\tilde{\mathbb{C}}_c^\#$ containing $\lambda_0$.

(b) The proof is the same as the proof of (a) except for two things. First, we consider the vector-valued functions $R_\lambda(A_n)\varphi$ and $R_\lambda(A)\varphi$. Secondly, since the map $T \to T^*$ is not #-continuous in the strong topology, one needs a separate argument to get from one half-plane of $*\tilde{\mathbb{C}}_c^\#$ to the other. Suppose that $\lambda_0$ is in the lower half-plane of $*\tilde{\mathbb{C}}_c^\#$. Then, as in (a), we get #-convergence everywhere in the lower half-plane of $*\tilde{\mathbb{C}}_c^\#$, in particular at $\lambda = -i$. The formula

$$(A_n - i)^{-1} - (A - i)^{-1} = \\ [(A_n + i)(A_n - i)^{-1}][(A_n + i)^{-1} - (A + i)^{-1}][(A_n + i)(A_n - i)^{-1}] \qquad (5.7.3)$$

which follows from elementary calculations, can then be used to prove that hyper infinite sequence $(A_n - i)^{-1}, n \in *\mathbb{N}$ #-converges strongly to $(A - i)^{-1}$. The above argument then shows that hyper infinite sequence $R_\lambda(A_n)$, $n \in *\mathbb{N}$ #-converges strongly to $R_\lambda(A)$ everywhere in the upper half-plane of $*\tilde{\mathbb{C}}_c^\#$.

For alternative ways of proving that strong #-convergence, $R_\lambda(A_n) \xrightarrow{s}_* R_\lambda(A)$ in one half-plane implies strong #-convergence in the other half-plane, see Theorem 5.7.9. We will investigate several aspects of generalized #-convergence. First, we ask how resolvent #-convergence is related to the #-convergence of other bounded functions of $A_n$ and $A$. Secondly, we investigate the relationship between the spectra of $A_n$ and the spectrum of $A$ if $A_n \to_* A$ in a generalized sense. Finally, we give criteria on the operators $A_n$, $A$ themselves which are sufficient to guarantee that $A_n \to_* A$ as $n \to *\infty$ in a generalized sense.

**Theorem 5.7.3.** Let $A_n$ and $A$ be self-#-adjoint operators.

(a) If $A_n \to_* A$ as $n \to *\infty$ in the #-norm resolvent sense and $f$ is a #-continuous function on $*\tilde{\mathbb{R}}_c^\#$ vanishing at $*\infty$, then $\|f(A_n) - f(A)\|_{\#} \to_* 0$ as $n \to *\infty$

(b) If $A_n \to_* A$ in the strong resolvent sense and $f$ is a bounded in $*\tilde{\mathbb{R}}_c^\#$ #-continuous function on $*\tilde{\mathbb{R}}_c^\#$, then $f(A_n)\varphi \to_* f(A)\varphi$ as $n \to *\infty$, for all $\varphi \in H^\#$.

**Proof** By the generalized Stone-Weierstrass theorem, polynomials in $(x + i)^{-1}$ and
$$ \left( \prod_{i=N_1+1}^{N_2} a^{\dagger}(f_i) \right) \left( \prod_{i=1}^{N_1} a^{\dagger}(f_i) \right) = \\ \mathrm{Ext} - \int_{*\mathbb{R}_c^{\#3N_2}} \left( \prod_{i=N_1+1}^{N_2} a^{\dagger}(p_i) \right) \left( \prod_{i=1}^{N_1} a^{\dagger}(-p_i) \right) \left( \prod_{i=1}^{N_2} f_i(p_i) \right) d^{\#}p_1 \dots d^{\#}p_{N_2} \qquad (53) $$

and

$$ N = \mathrm{Ext} - \int_{*\mathbb{R}_c^{\#3}} a^{\dagger}(p) a(p) d^{\#}p \qquad (54) $$

The generator of time translations in the free scalar field theory of mass $m$ is given by

$$ \mathbf{H}_0 = \mathrm{Ext} - \int_{*\mathbb{R}_c^{\#3}} \mu(p) a^{\dagger}(p) a(p) d^{\#}p \qquad (54) $$

$\mathbf{H}_0$ is called the free Hamiltonian of mass $m$. (52), (53), and (54) involve no formal manipulations, but are mathematical statements about quadratic forms.

**Theorem X.44** Let $n_1$ and $n_2$ be nonnegative integers and suppose that $W \in \mathcal{L}_2^{\#}(\mathbb{R}_c^{\#3(n_1+n_2)})$. Then there is a unique operator $T_W$ on $\mathcal{F}_s(\mathcal{L}_2^{\#}(\mathbb{R}_c^{\#3}))$ so that $D_{\mathcal{L}_{\mathrm{fin}}^{\#}} \subset D(T_W)$ is a core for $T_W$ and

(a) as $*\mathbb{C}_c^{\#}$-valued quadratic forms on $D_{\mathcal{L}_{\mathrm{fin}}^{\#}} \times D_{\mathcal{L}_{\mathrm{fin}}^{\#}}$

$$ T_W = \mathrm{Ext} - \int_{*\mathbb{R}_c^{\#3(n_1+n_2)}} W(k_1, \dots, k_{n_1}, p_1, \dots, p_{n_2}) \left( \prod_{i=1}^{n_1} a^{\dagger}(k_i) \right) \left( \prod_{i=1}^{n_2} a(p_i) \right) d^{\#n_1}kd^{\#n_2}p \qquad (55) $$

(b) If $m_1$ and $m_2$ are nonnegative integers so that $m_1 + m_2 = n_1 + n_2$, then $(1+N)^{-m_1/2}T_W(1+N)^{-m_2/2}$ is a bounded operator with

$$ \|(1+N)^{-m_1/2} T_W(1+N)^{-m_2/2} \| < C(m_1,m_2) \|W\|_{\mathcal{L}_2^{\#}}. \qquad (56) $$

In particular, if $m_1 = n_1$ and $m_2 = n_2$, then

$$ \|(1+N)^{-n_1/2} T_W(1+N)^{-n_2/2} \| < C(m_1,m_2) \|W\|_{\mathcal{L}_2^{\#}}. \qquad (57) $$

(c) As $*\mathbb{C}_c^{\#}$-valued quadratic forms on $D_{\mathcal{L}_{\mathrm{fin}}^{\#}} \times D_{\mathcal{L}_{\mathrm{fin}}^{\#}}$

$$ T_W^* = \mathrm{Ext} - \int_{*\mathbb{R}_c^{\#3(n_1+n_2)}} \overline{W(k_1, \dots, k_{n_1}, p_1, \dots, p_{n_2})} \left( \prod_{i=1}^{n_2} a^{\dagger}(k_i) \right) \left( \prod_{i=1}^{n_1} a(p_i) \right) d^{\#n_1}kd^{\#n_2}p \qquad (58) $$

(d) If $W_n \to_W W$ in $\mathcal{L}_2^{\#}(\mathbb{R}_c^{\#3(n_1+n_2)})$, then $T_{W_n} \to_W T_W$ strongly on $D_{\mathcal{L}_{\mathrm{fin}}^{\#}}$.

(e) $F_0$ is contained in $D(T_W)$ and $D(T_W^*)$, and on vectors in $F_0$, $T_W$ and $T_W^*$ are given by the explicit formulas

$$ (T_W \psi)^{(l-n_2+n_1)} = K(l, n_1, n_2) \mathbf{S} \times \left[ \mathrm{Ext} - \int_{*\mathbb{R}_c^{\#3n_2}} W(k_1, \dots, k_{n_1}, p_1, \dots, p_{n_2}) \psi^{(l)}(p_1, \dots, p_{n_2}, k_{n_1+1}, \dots, k_{n_1+l-n_2}) d^{\#n_2}p \right] \qquad (59) $$

$$ (T_W \psi)^n = 0 \text{ if } n < n_1 - n_2 $$

$$ (T_W^* \psi)^{(l-n_1+n_2)} = K(l, n_2, n_1) \mathbf{S} \times \left[ \mathrm{Ext} - \int_{*\mathbb{R}_c^{\#3(n_1)}} \overline{W(k_1, \dots, k_{n_1}, p_1, \dots, p_{n_2})} \psi^{(l)}(k_1, \dots, k_{n_1}, p_{n_2+1}, \dots, p_{n_2+l-n_1}) d^{\#n_1}k \right] \qquad (60) $$

$$ (T_W^* \psi)^n = 0 \text{ if } n < n_2 - n_1 \text{ where } \mathbf{S} \text{ is the symmetrization operator and} $$
$(x - i)^{-1}$ are $\#$-dense in $C^{*\infty}(\tilde{\mathbb{R}}_c^\#)$, the $\#$-continuous functions vanishing at hyper infinity. Thus, given $\varepsilon \approx 0, \varepsilon > 0$, we can find an hyperfinite polynomial $P(s, t)$ so that

$$ \|f(x) - P((x+i)^{-1}, (x-i)^{-1})\|_{\infty} \le \frac{\varepsilon}{3}. \qquad (5.7.4) $$

Therefore,

$$ \|f(A_n) - P((A_n + i)^{-1}, (A_n - i)^{-1})\|_{\infty} \le \frac{\varepsilon}{3} \qquad (5.7.5) $$

and

$$ \|f(A) - P((A + i)^{-1}, (A - i)^{-1})\|_{\infty} \le \frac{\varepsilon}{3}. \qquad (5.7.6) $$

If $A_n \to_A A$ as $n \to \infty$ in the $\#$-norm resolvent sense, then

$$ P((A_n + i)^{-1}, (A_n - i)^{-1}) \to_A P((A + i)^{-1}, (A - i)^{-1}) \qquad (5.7.7) $$

in $\#$-norm as $n \to \infty$, and thus for hyperfinite $n$ large enough, $\|f(A_n) - f(A)\|_{\#} \le \varepsilon$. This proves (a).

To prove (b) we first note that the same proof as above shows that if $A_n \to_A A$ in the strong resolvent sense and $h \in C^{*\infty}(\tilde{\mathbb{R}}_c^\#)$, then $h(A_n)\varphi \to_A h(A)\varphi$. Let $\psi \in H^\#$ and $\varepsilon \approx 0, \varepsilon > 0$ be given and define $g_m(x) = \text{Ext}\text{-exp}(-x^2/m)$. Since $g_m(x) \to_A 1$ pointwise, $g_m(A)\psi \to_A \psi$ by spectral theorem, so we can find an $m$ with $\|g_m(A)\psi - \psi\|_{\#} \le \varepsilon(6\|f\|_{\infty})^{-1}$. Furthermore since $g_m \in C^{*\infty}(\tilde{\mathbb{R}}_c^\#)$, $g_m(A_n)\psi \to_A g_m(A)\psi$ by the remark above, so we can find an $N_0$, so that $n > N_0$ implies

$$ \|g_m(A_n)\psi - g_m(A)\psi\|_{\#} \le \varepsilon(6\|f\|_{\infty})^{-1}. \quad \text{Therefore, if } n \ge N_0, $$

$$ \|g_m(A_n)\psi - \psi\|_{\#} \le \varepsilon(3\|f\|_{\infty})^{-1}. \qquad (5.7.8) $$

Since $fg_m$ is $\#$-continuous and goes to zero at $\infty$, there is an $N_1$ so that $n \ge N_1$ implies

$$ \|f(A_n)g_m(A_n)\psi - f(A)g_m(A)\psi\|_{\#} \le \frac{\varepsilon}{3}. \qquad (5.7.9) $$

Let $N = \max(N_0, N_1)$. Then for $n \ge N$,

$$ \|f(A_n)\psi - f(A)\psi\|_{\#} \le \|f(A_n)g_m(A_n)\psi - f(A)g_m(A)\psi\|_{\#} + \|A_n\|_{\#}\|g_m(A_n)\psi - \psi\|_{\#} + \|A\|_{\#}\|g_m(A)\psi - \psi\|_{\#}. \qquad (5.7.10) $$

Since $\psi$ and $\varepsilon$ were arbitrary, this proves (b).

As an example of an application of part (a) let $(A_n)_{n=1}^\infty$ and $A$ be positive self-$\#$-adjoint operators. Then, if $A_n \to_A A$ in the $\#$-norm resolvent sense $\text{Ext}\text{-exp}(-tA_n)$ $\#$-converges in $\#$-norm to $\text{Ext}\text{-exp}(-tA)$ for each positive $t$. To see that part (a) does not extend to all of $C^\#(\tilde{\mathbb{R}}_c^\#)$, notice that on $L_2^\#(\tilde{\mathbb{R}}_c^\#)$ the operators $A_n = (1 - n^{-1})x$ $\#$-converge to the operator $A = x$ in the $\#$-norm resolvent sense but $\|\text{Ext}\text{-exp}(iA_n) - \text{Ext}\text{-exp}(iA_n)\|_{\#} = 1$ for all $n \in \mathbb{N}$.

An important application of part (b) is the following generalization of the classical Trotter theorem.

**Theorem 5.7.4.** Let $(A_n)_{n=1}^\infty$ and $A$ be self-$\#$-adjoint operators. Then $A_n \to_A A$ in the strong resolvent sense if and only if $\text{Ext}\text{-exp}(itA_n)$ $\#$-converges strongly to $\text{Ext}\text{-exp}(itA)$ for each $t$.

**Proof** Since $\text{Ext}\text{-exp}(itx)$ is a bounded $\#$-continuous function of $x$, Theorem 5.7.3 implies that if $A_n \to_A A$ in the strong resolvent sense, then $\text{Ext}\text{-exp}(itA_n) \to_A \text{Ext}\text{-exp}(itA)$ as $n \to \infty$, strongly for each $t$.
$$K(l, n_1, n_2) = \left[ \frac{l!(l + n_1 - n_2)!}{((l - n_2)!)^2} \right]^{1/2}. \qquad (61)$$

**Proof.** For vectors in $D_{\mathcal{L}_{\text{fin}}^{\#}}$, we define $T_W\psi$ by the formula (X.82a). By the Schwarz inequality and the fact that S is a projection,

$$\|(T_W\psi)^{(l-n_2+n_1)}\|^2 \leq K(l, n_1, n_2) \|\psi^{(l)}\|^2 \|W\|^2. \qquad (62)$$

If we now define an operator $T_W^*\psi$, on $D_{\mathcal{L}_{\text{fin}}^{\#}}$ by using the formula in (62), then for all $\varphi$ and $\psi$ in $D_{\mathcal{L}_{\text{fin}}^{\#}}$ one easily verifies that $(\varphi, T_W\psi) = (T_W^*\varphi, \psi)$. Thus, $T_W$ is #-closable and $T_W^*$ is the restriction of the adjoint of $T_W$ to $D_{\mathcal{L}_{\text{fin}}^{\#}}$. From now on we will use $T_W$ to denote $\bar{T}_W$ and $T_W^*$ to denote the adjoint of $T_W$. By the definition of $T_W$, $D_{\mathcal{L}_{\text{fin}}^{\#}}$ is a #-core and further, since $T_W$ is bounded on the $l$-particle vectors in $D_{\mathcal{L}_{\text{fin}}^{\#}}$, we have $F_0 \subset D(T_W)$. Since the right-hand side of (59) is also bounded on the $l$-particle vectors, (X.82a) represents $T_W$ on all $l$-particle vectors. The proof of the statements in (e) about $T_W^*$ are the same. To prove (b), let $\psi \in D_{\mathcal{L}_{\text{fin}}^{\#}}$. Then by the above computation

$$\left\| \left( (1+N)^{-m_1/2} T_W (1+N)^{-m_2/2} \psi \right)^{(l-n_2+n_1)} \right\|^2 \le \left[ \frac{K(l, n_1, n_2)}{(1+l-n_2+n_1)^{m_1/2} (1+l)^{m_2/2}} \right]^2 \|\psi^{(l)}\|^2 \|W\|^2 \qquad (63)$$

so that

$$\left\| \left( (1+N)^{-m_1/2} T_W (1+N)^{-m_2/2} \psi \right)^{(l-n_2+n_1)} \right\| \le \left[ \sup_{l \in \mathbb{N}} \frac{K(l, n_1, n_2)}{(1+l-n_2+n_1)^{m_1/2} (1+l)^{m_2/2}} \right] \|\psi^{(l)}\| \|W\| \le C(m_1, m_2) \|\psi^{(l)}\| \|W\| \qquad (64)$$

where

$$C(m_1, m_2) = \sup_{l \in \mathbb{N}} \frac{K(l, n_1, n_2)}{(1+l-n_2+n_1)^{m_1/2} (1+l)^{m_2/2}} < \infty^{\#} \qquad (65)$$

since $m_1 + m_2 = n_1 + n_2$. In all the sup's only $l$ so that $l - n_2 + n_1 > 0$ occur since the other terms are annihilated by the action of $T_W$. Thus, $(1+N)^{-m_1/2}T_W(1+N)^{-m_2/2}$ extends to a hyper bounded operator on $\mathcal{F}_s(H^\#)$ with norm less than or equal to $C(m_1, m_2)$. If $m_1 = n_1$ and $m_2 = n_2$, then $C(m_1, m_2) = 1$.

To prove (d) we need only note that if $\psi = (0, \dots, \psi^{(l)}, 0, \dots) \in D_{\mathcal{L}_{\text{fin}}^{\#}}$ and $W_n \to_{\#} W$ in $\mathcal{L}_2^{\#}$, then

$$\|T_{W_n}\psi - T_W\psi\| = \|(T_{W_n-W})\psi\| \leq K(l, n_1, n_2) \|W_n - W\| \|\psi\|, \qquad (66)$$

where $\#\lim_{n \to \infty} K(l, n_1, n_2) \|W_n - W\| \|\psi\| = 0$.

Since $D_{\mathcal{L}_{\text{fin}}^{\#}}$ consists of finite linear combinations of such vectors, we have shown that $T_{W_n}$ #-converges strongly on $D_{\mathcal{L}_{\text{fin}}^{\#}}$ to $T_W$ if $W_n \to_{\#} W$ in $\mathcal{L}_2^{\#}$.

To prove (a) let $\psi_1, \psi_2 \in D_{\mathcal{L}_{\text{fin}}^{\#}}$ with $\psi_1 = (0, \dots, \psi^{(l-n_2+n_1)}, 0, \dots)$ and $\psi_1 = (0, \dots, \psi^{(l)}, 0, \dots)$.

Then, if $W = (\prod_{i=1}^{n_1} f_i(k_i)) (\prod_{i=1}^{n_2} g_i(k_i))$ the definition of the form
To prove the theorem in the other direction, we first derive a formula for the resolvent of a self-#-adjoint operator A. Suppose that $\mathrm{Im}\,\mu < 0$. Then, by the functional calculus

$$ \begin{aligned} \langle \psi, R_{\mu}(A)\varphi \rangle_{\#} &= \mathrm{Ext} - \int_{\ast\tilde{\mathbb{R}}_{c}^{\#}} \left( \frac{1}{\mu - \lambda} \right) d^{\#} \langle \psi, P_{\lambda}\varphi \rangle_{\#} = \\ & \mathrm{Ext} - \int_{\ast\tilde{\mathbb{R}}_{c}^{\#}} \left( \mathrm{Ext} - \int_{0}^{\infty} i[\mathrm{Ext}\,-\exp(-it\mu)][\mathrm{Ext}\,-\exp(it\lambda)]d^{\#}t \right) = \\ & \mathrm{Ext} - \int_{0}^{\infty} i[\mathrm{Ext}\,-\exp(-it\mu)]^{\#} \langle \psi, \mathrm{Ext}\,-\exp(itA)\varphi \rangle_{\#} = \\ & \left\langle \psi, \mathrm{Ext} - \int_{0}^{\infty} i[\mathrm{Ext}\,-\exp(-it\mu)][\mathrm{Ext}\,-\exp(itA)]\varphi d^{\#}t \right\rangle_{\#} \end{aligned} \quad (5.7.11) $$

Therefore,

$$ R_{\mu}(A)\varphi = \mathrm{Ext} - \int_{0}^{\infty} i[\mathrm{Ext}\,-\exp(-it\mu)][\mathrm{Ext}\,-\exp(itA)]\varphi d^{\#}t \quad (4.7.12) $$

where the #-integral is a Riemann #-integral. The third step in the computation uses generalized Fubini's theorem. Applying (5.7.12) to the operators $A_n$ and A we obtain

$$ \| R_{\mu}(A_n)\varphi - R_{\mu}(A)\varphi \|_{\#} \le \mathrm{Ext} - \int_{0}^{\infty} [\mathrm{Ext}\,-\exp(t\mathrm{Im}\,\mu)] \|[\mathrm{Ext}\,-\exp(itA_n) - \mathrm{Ext}\,-\exp(itA)]\varphi \|_{\#} d^{\#}t \quad (5.7.13) $$

so if $\mathrm{Ext}\,-\exp(itA_n) \to_{\#} \mathrm{Ext}\,-\exp(itA)$ as $n \to \infty$ for each $t$, $\|R_{\mu}(A_n)\varphi - R_{\mu}(A)\varphi\|_{\#} \to_{\#} 0$ as $n \to \infty$ by the generalized Lebesgue dominated convergence theorem. Using a formula similar to (5.7.12) one concludes in the same way that $\|R_{\mu}(A_n)\varphi - R_{\mu}(A)\varphi\|_{\#} \to_{\#} 0$ for $\mathrm{Im}\,\mu > 0$. We remark that it is possible to show that if $A_n \to_{\#} A$ in the strong resolvent sense, then $\mathrm{Ext}\,-\exp(itA_n) \to_{\#} \mathrm{Ext}\,-\exp(itA)$ for each $\varphi$ uniformly for $t$ in any hyperfinite interval.

**Theorem 5.7.5.** (Generalized Trotter-Kato theorem) Let $(A_n)_{n=1}^{\infty}$ be a sequence of self-#-adjoint operators. Suppose that there exist points, $\lambda_0$ in the upper half-plane of $\ast\tilde{\mathbb{C}}_c^\#$ and $\mu_0$ in the lower half-plane of $\ast\mathbb{C}_c^\#$ so that $R_{\lambda_0}(A_n)\varphi$ and $R_{\mu_0}(A_n)\varphi$ #-converge as $n \to \infty$ for each $\varphi \in H^\#$. Suppose further that one of the limiting operators, $T_{\lambda_0}$ or $T_{\mu_0}$, has a #-dense range. Then there exists a self-#-adjoint operator A so that $A_n \to_{\#} A$ as $n \to \infty$ in the strong resolvent sense.

**Proof** Since $\|R_{\lambda_0}(A_n)\|_{\#} \le |\mathrm{Im}\,\lambda_0|^{-1}$, $\|T_{\lambda_0}\| \le |\mathrm{Im}\,\lambda_0|^{-1}$, and so

$$ T_{\lambda} = \mathrm{Ext} - \sum_{n=0}^{\infty} (\lambda_{0} - \lambda)^{n} (T_{\lambda_{0}})^{n+1}. \quad (5.7.14) $$

is well defined for $|\lambda_0 - \lambda| \le |\mathrm{Im}\,\lambda_0|^{-1}$. Furthermore, since $R_{\lambda_0}(A_n)\varphi \to_{\#} T_{\lambda_0}\varphi$, $R_{\lambda}(A_n)\varphi \to_{\#} T_{\lambda}\varphi$ in the same circle.

Continuing in this way we can define an #-analytic operator valued function $T_{\lambda}$ in the half-plane containing $\lambda_0$ which is the strong #-limit of $R_{\lambda}(A_n)$. Since the half-plane is
$(\prod_{i=1}^{n_1} a^{\dagger}(k_i))(\prod_{i=1}^{n_2} a_i(k_i))$ shows that

$$ (\psi_1, T_W \psi_2) = \mathrm{Ext} - \int_{\mathbb{R}_c^{\#3n_2}} W(k_1, \dots, k_{n_1}, p_1, \dots, p_{n_2}) \times \\ (\psi_1, (\prod_{i=1}^{n_1} a^{\dagger}(k_i))(\prod_{i=1}^{n_2} a_i(k_i)) \psi_2) d^{\#n_1} k d^{\#n_2} p \qquad (67) $$

Since both sides of (X.83) are linear in $W$, the relationship continues to hold for the all such $W$'s that are hyperfinite linear combinations of such products. Since

$$ (\psi_1, (\prod_{i=1}^{n_1} a^{\dagger}(k_i))(\prod_{i=1}^{n_2} a_i(k_i)) \psi_2) \in \mathcal{L}_2^{\#}(\mathbb{R}_c^{\#3(n_1+n_2)}) \qquad (68) $$

and since (d) holds, both the right- and left-hand sides of (X.83) are continuous linear functionals on $\mathbb{R}_c^{\#3(n_1+n_2)}$. Since they agree on a #-dense set, they agree everywhere. Finally, (68) extends by linearity to all of $D_{\mathcal{L}_{\mathrm{fin}}^{\#}} \times D_{\mathcal{L}_{\mathrm{fin}}^{\#}}$.

This proves (a); the proof of (c) is similar. |

Finally, we note that as quadratic forms on $D_{\mathcal{L}_{\mathrm{fin}}^{\#}}$ we can express the free scalar field and the time zero fields in terms of $a^{\dagger}(k)$ and $a(k)$:

$$ \Phi_{m,x}(x,t) = $$

$$ \frac{1}{(2\pi\#)^{3/2}} \int_{|p|\le x} \{[\mathrm{Ext}\text{-exp}(\mu(p)t - ipx)]a^{\dagger}(p) + [\mathrm{Ext}\text{-exp}(-\mu(p)t + ipx)]a(p)\} \frac{d^{\#3}p}{\sqrt{2\mu(p)}} \qquad (69) $$

$$ \varphi_{m,x}^{\#}(x) = \frac{1}{(2\pi\#)^{3/2}} \int_{|p|\le x} \{[\mathrm{Ext}\text{-exp}(-ipx)]a^{\dagger}(p) + [\mathrm{Ext}\text{-exp}(ipx)]a(p)\} \frac{d^{\#3}p}{\sqrt{2\mu(p)}} \qquad (70) $$

$$ \pi_{m,x}^{\#}(x) = \frac{1}{(2\pi\#)^{3/2}} \int_{|p|\le x} \{[\mathrm{Ext}\text{-exp}(-ipx)]a^{\dagger}(p) - [\mathrm{Ext}\text{-exp}(ipx)]a(p)\} \sqrt{\frac{\mu(p)}{2}} d^{\#3}p. \qquad (71) $$

## 2. $Q^\#$-space representation of the Fock space structures

In this section the construction of $Q^\#$-space and $L_2^\#(Q^\#, d^\#\mu)$, another representation of the Fock space structures are presented. In analogy with the one degree of freedom case where $\mathcal{F}^\#(*\mathbb{R}_c^\#)$ is isomorphic to $L_2^\#(*\mathbb{R}_c^\#, d^\#x)$ in such a way that $\Phi_S(1)$ becomes multiplication by $x$, we will construct a #-measure space $\langle Q^\#, \mu^\# \rangle$, with $\mu(Q^\#) = 1$, and a unitary map $S: \mathcal{F}_S^\#(*\mathbb{R}_c^\#) \to L_2^\#(Q^\#, d^\#\mu)$ so that for each $f \in \mathbf{H}_c^\#$, $S\varphi^\#(f)S^{-1}$ acts on $L_2^\#(Q, d^\#\mu^\#)$ by multiplication by a #-measurable function. We can then

show that in the case of the free scalar field of mass $m$ in 4-dimensional space-time, $V = SH_I(g)S^{-1}$ is just multiplication by a function $V(q)$ which is in $L_p^\#(Q, d^\#\mu)$ for each $p \in \mathbb{N}^\#$. Let $\{f_n\}_{n=1}^{\infty}$ be an orthonormal basis for $\mathbf{H}^\#$ so that each $f_n \in \mathbf{H}_c^\#$ and let $\{g_k\}_{k=1}^N, N \in \mathbb{N}^\#$ be a finite or hyperfinite subcollection of the $\{f_n\}_{n=1}^{\infty}$. Let $\mathbf{P}_N$ be a set of the all external hyperfinite polynomials $\mathrm{Ext}\text{-P}[u_1, \dots, u_N]$ and $\mathcal{F}_N^\#$ be the #-closure of the set

$$ \{\mathrm{Ext}\text{-P}[\varphi^\#(g_1), \dots, \varphi^\#(g_N)]\}_{P \in \mathbf{P}_N} \qquad (1) $$

in $\mathcal{F}_S^\#(\mathbf{H}^\#)$ and define $F_0^N = \mathcal{F}_N^\# \cap F_0$. From Theorem X.43 (and its proof) it follows that $\varphi^\#(g_k)$ and $\pi^\#(g_l)$, for all $1 \le k, l \le N$ are essentially self-adjoint on $F_0^N$ and that
simply #-connected, the determination of $T_λ$ at a point $λ$ is independent of the path taken from $λ_0$ to $λ$. The same argument for the half-plane containing shows that we can extend the definition of $T_λ$ to that half-plane of *$\tilde{\mathbb{C}}_c^\#$ so that for all $λ$ with $\mathrm{Im}\,λ \neq 0$

$$ T_{\lambda}\varphi = \#\lim_{n \to \infty} R_{\lambda}(A_n)\varphi. \qquad (5.7.15) $$

The $T_λ$ commute, satisfy the first resolvent equation, and $T_λ^* = T_λ$ since these statements are true for each $R_λ(A_n)$ It follows from the first resolvent formula and the commutativity that the ranges of all the $T_λ$ are equal; we denote this common range by $D$. By hypothesis, $D$ is #-dense and this implies that the kernel of $T_λ$ is empty, since $\mathrm{Ker}(T_λ) = (\mathrm{Ran}(T_λ^*))^\top = (\mathrm{Ran}(T_λ))^\top = D^\top = \{0\}$. We can therefore define $A = \lambda I - T_λ^{-1}$ on $D$ and a short calculation with the resolvent equation shows that this definition is independent of which $λ$ with $\mathrm{Im}\,λ \neq 0$, is chosen. Since $\mathrm{Ran}(A \pm i) = \mathrm{Ran}(T_{\pm i}^{-1})$ operator $A$ is self-#-adjoint. It is clear that the resolvent of $A$ is $T_λ$.

Notice that in the Trotter-Kato theorem we need convergence at two points, one in the upper half-plane and one in the lower half-plane of *$\tilde{\mathbb{C}}_c^\#$*. For, we cannot use Theorem 5.7.3 until we know that the #-limiting operator is self-#-adjoint, and the self-#-adjointness proof depends on the #-convergence in both half-planes of *$\tilde{\mathbb{C}}_c^\#$*. The Trotter-Kato theorem is important since its hypotheses do not assume the a priori existence of a #-limiting operator $A$. It can be used to assert the existence of a generalized #-limit of a sequence of self-#-adjoint operators. This can also be done with the one-parameter groups. To see why it is necessary to use the resolvents or groups rather than the operators themselves to prove such an existence theorem consider the following example: Let $A$ be a closed symmetric operator which is not self-#-adjoint but which has a self-#-adjoint extension $\tilde{A}$. Let $P_n$ be the spectral projection of $\tilde{A}$ corresponding to the interval $[-n, n]$. Then $P_n \tilde{A} P_n$ are bounded self-#-adjoint operators (and therefore essentially self-#-adjoint on $D(A)$) such that for all $\varphi \in D(A) : P_n \tilde{A} P_n \varphi \to_\# \tilde{A} \varphi = A \varphi$. Thus the $P_n A P_n$ are essentially self-#-adjoint on $D(A)$ and the strong #-limit exists but the #-limit is not essentially self-#-adjoint. One of the most useful aspects of generalized #-convergence is that the spectra and projections of the $A_n$ are related to the spectrum and projections of $A$.

**Theorem 5.7.6.** Let $(A_n)_{n=1}^\infty$ and $A$ be self-#-adjoint operators and suppose that $A_n \to_\# A$ in the #-norm resolvent sense. Then

(a) If $\mu \notin \sigma(A)$, then $\mu \notin \sigma(A_n)$ for $n \in *\mathbb{N}$ sufficiently large and

$$ \|R_{\mu}(A_n) - R_{\mu}(A)\|_{\#} \to_{\#} 0 \qquad (5.7.16) $$

(b) Let $a, b \in *\tilde{\mathbb{R}}_c^\#$, $a < b$, and suppose that $a \in \rho(A)$, $b \in \rho(A)$. Then

$$ \|P_{(a,b)}(A_n) - P_{(a,b)}(A)\|_{\#} \to_{\#} 0 \qquad (5.7.17) $$

**Proof** (a) We need only consider the case where $\mu \in *\tilde{\mathbb{R}}_c^\#$. Since $\mu \in \rho(A)$, there is a $\delta \approx 0, \delta > 0$ so that $(\mu - \delta, \mu + \delta) \cap \sigma(A) = \emptyset$. Thus, by the functional calculus, $\|R_{\mu+i\delta/3}(A)\|_{\#} < 1/\delta$. Now, we can find $N$ so that $\|R_{\mu+i\delta/3}(A_n)\|_{\#} < 2/\delta$ for $n \ge N$ which implies that the power series for $R_λ(A_n)$ about $\mu + i\delta/3$ has radius of #-convergence at least $5/2$. We already know that where the series #-converges it is an inverse for $A_n$. So, $\mu \in \rho(A_n)$ for $n \ge N$ and $\|R_\mu(A_n) - R_\mu(A)\|_{\#} \to_\# 0$ as $n \to *\infty$.

To prove (b), we note that since $a, b \in \rho(A)$, there exists $\varepsilon < (1/2)(b-a)$ and an $N$,
$$ (\mathrm{Ext}-\exp[\mathrm{it}\varphi^{\#}(g_k)])(\mathrm{Ext}-\exp[\mathrm{is}\pi^{\#}(g_l)]) = \\ (\mathrm{Ext}-\exp[-\mathrm{ist}\delta_{kl}])(\mathrm{Ext}-\exp[\mathrm{is}\pi^{\#}(g_l)])(\mathrm{Ext}-\exp[\mathrm{it}\varphi^{\#}(g_k)]). \qquad (2) $$

Thus we have a representation of the generalized Weyl relations in which the vector $\Omega_0$ satisfies $([\varphi^{\#}(g_k)]^2 + [\pi^{\#}(g_k)]^2 - 1)\Omega_0 = 0$ and is #-cyclic for the operators $\{\varphi^{\#}(g_k)\}_{k=1}^N, N \in \mathbb{N}^#$. Therefore there is a unitary map $\tilde{S}^{(N)} : \mathcal{F}_N^{\#} \to L_2^{\#}(\mathbb{R}_c^{\#N})$ so that

$$ \begin{aligned} \tilde{S}^{(N)} \varphi^{\#}(g_k) (\tilde{S}^{(N)})^{-1} &= x_k \\ \tilde{S}^{(N)} \pi^{\#}(g_k) (\tilde{S}^{(N)})^{-1} &= \frac{1}{i} \frac{d^{\#}}{dx_k^{\#}} \end{aligned} \qquad (3) $$

and

$$ \tilde{S}^{(N)} \mathbf{Q}_0 = \pi_{\#}^{-N/4} \left\{ \mathrm{Ext} \cdot \exp \left[ - \left( \mathrm{Ext} \cdot \sum_{k=1}^{N} \frac{x_k^2}{2} \right) \right] \right\}. \qquad (4) $$

It is convenient to use the Hilbert space

$$ L_2^{\#} ( * \mathbb{R}_c^{\#N}, \pi_{\#}^{-N/2} d^{\#N} x \{ \mathrm{Ext} \cdot \exp \left[ - \left( \mathrm{Ext} \cdot \sum_{k=1}^{N} \frac{x_k^2}{2} \right) \right] \} ) $$

instead of $L_2^{\#}(\mathbb{R}_c^{\#N})$ so let $d^{\#}\mu_k = \pi_{\#}^{-1/2} \exp(-x_k^2/2) d^{\#}x_k$ and define

$$ (Tf)(x) = \pi_{\#}^{N/4} \left[ \mathrm{Ext} \cdot \exp \left( \mathrm{Ext} \cdot \sum_{k=1}^{N} \frac{x_k^2}{2} \right) \right] f(x). \qquad (5) $$

Then $T$ is a unitary map of $L_2^{\#}(\mathbb{R}_c^{\#N})$ onto $L_2^{\#}(\mathbb{R}_c^{\#N}, \mathrm{Ext} \cdot \prod_{k=1}^N d^{\#}\mu_k^{\#})$ and if we let $S^{(N)} = T\tilde{S}^{(N)}$ we get

$$ \begin{aligned} S^{(N)} : \mathcal{F}_N^{\#} &\to L_2^{\#} ( * \mathbb{R}_c^{\#N}, \mathrm{Ext} \cdot \prod_{k=1}^N d^{\#} \mu_k^{\#} ), \\ S^{(N)} \varphi^{\#}(g_k) (S^{(N)})^{-1} &= x_k, \\ S^{(N)} \pi^{\#}(g_k) (S^{(N)})^{-1} &= -\frac{x_k}{i} + \frac{1}{i} \frac{d^{\#}}{d^{\#}x_k}, \\ S^{(N)} \mathbf{Q}_0 &= 1, \end{aligned} \qquad (6) $$

where $1$ is the function identically one. Note that each $\mu_k^{\#}$ has mass one, which implies that

$$ \begin{aligned} \langle \mathbf{Q}_0, (\mathrm{Ext} \cdot \prod_{k=1}^N P_k[\varphi^{\#}(g_k)]) \mathbf{Q}_0 \rangle &= \\ \int_{*\mathbb{R}_c^{\#N}} (\mathrm{Ext} \cdot \prod_{k=1}^N P_k[x_k]) (\mathrm{Ext} \cdot \prod_{k=1}^N d^{\#} \mu_k^{\#}) &= \\ \mathrm{Ext} \cdot \prod_{k=1}^N \int_{*\mathbb{R}_c^{\#}} P[x_k] d^{\#} \mu_k^{\#} &= \mathrm{Ext} \cdot \prod_{k=1}^N \int_{*\mathbb{R}_c^{\#}} \langle \mathbf{Q}_0, P_k[\varphi^{\#}(g_k)] \mathbf{Q}_0 \rangle, \end{aligned} \qquad (7) $$

where $P_1, \dots, P_N$ are external hyperfinite polynomials. This formula (7) can also be proven by direct computations on $\mathcal{F}_s^{\#}(\mathbf{H}^{\#})$.

Now it is easy to see how to construct $\langle Q^{\#}, d^{\#}\mu^{\#} \rangle$. We define $Q^{\#} = \times_{k=1}^{\infty} * \mathbb{R}_c^{\#}$. Take the $\sigma^-$-algebra generated by hyper infinite products of #-measurable sets in $*\mathbb{R}_c^{\#}$ and set $\mu^{\#} = \otimes_{k=1}^{\infty} \mu_k^{\#}$. We denote the points of $Q^{\#}$ by $q = \langle q_1, q_2, \dots \rangle$. Then $\langle Q^{\#}, d^{\#}\mu^{\#} \rangle$ is a #-measure space and the set of functions of the form $P(q_1, q_2, \dots)$, where $P$ is a
so that $\sup_{n \ge N} \{(\|(A_n - a)^{-1}\|_#, \|(A_n - b)^{-1}\|_#)\} \le 1/\epsilon$. Therefore, by the functional calculus, $\sigma(A_n) \cap (a,b) \subset (a+\epsilon, b-\epsilon)$ for $n \ge N$. Let $f_\epsilon$ be a #-continuous function which equals one on $(a+\epsilon, b-\epsilon)$ and is equal to zero outside $(a,b)$. Then $P_{(a,b)}(A_n) = f_\epsilon(A_n)$ and $P_{(a,b)}(A) = f_\epsilon(A)$ and so by Theorem 5.7.3 one obtains $\|P_{(a,b)}(A_n) - P_{(a,b)}(A)\|_# \to_# 0$ as $n \to *\infty$.

**Theorem 5.7.7.** Let $(A_n)_{n=1}^{*\infty}$ and $A$ be self-#-adjoint operators and suppose that $A_n \to_* A$ in the strong resolvent sense. Then

(a) If $a,b \in *\tilde{\mathbb{R}}_c^\# a < b$, and $(a,b) \cap \sigma(A_n) = \emptyset$ for all $n \in *\mathbb{N}$, then $(a,b) \cap \sigma(A) = \emptyset$. That is, if $\lambda \in \sigma(A)$, then there exists $\lambda_n \in \sigma(A_n)$ so that $\lambda_n \to_* \lambda$.

(b) If $a,b \in *\tilde{\mathbb{R}}_c^\# a < b$, and $a,b \notin \sigma_{pp}(A)$ then $P_{(a,b)}(A_n)\varphi \to_* P_{(a,b)}(A)\varphi$ for all $\varphi \in H^\#$.

**Proof** By the functional calculus, the statement that $(a,b) \cap \sigma(A_n) = \emptyset$ is equivalent to the statement that $\|(A - \lambda_0)^{-1}\|_# \le \sqrt{2}/(b-a)$ where $\lambda_0 = (a+b)/2 + i(b-a)/2$. But $(A_n - \lambda_0)^{-1}$ #-converges strongly to $(A - \lambda_0)^{-1}$ so we have $\|(A - \lambda_0)^{-1}\|_# \le *\overline{\lim}_{n \to *\infty} \|(A_n - \lambda_0)^{-1}\|_# \le \sqrt{2}/(b-a)$. This proves (a).

To prove (b), we find uniformly bounded sequences of #-continuous functions $(f_n)_{n=1}^{*\infty}$ and $(g_n)_{n=1}^{*\infty}$ so that $0 \le f_n \le \chi_{(a,b)}$, $f_n(x) \to_* \chi_{(a,b)}(x)$ pointwise and $\chi_{(a,b)} \le g_n$, $g_n(x) \to_* \chi_{[a,b]}(x)$ pointwise. Then $f_n(A) \to_* P_{(a,b)}(A)$ and $g_n(A) \to_* P_{[a,b]}(A)$ strongly. Since $a,b \notin \sigma_{pp}(A)$, $P_{(a,b)}(A) = P_{[a,b]}(A)$ which means that given $\psi$ and $\epsilon \approx 0, \epsilon > 0$, we can find #-continuous functions $f,g$, with $f \le \chi_{(a,b)} \le \chi_{[a,b]} \le g$ so that $\|f(A)\psi - f(A)\psi\|_# \le \epsilon/5$. By Theorem 5.7.3(b) we can find $N \in *\mathbb{N}$ so that $n > N$ implies $\|f(A_n)\psi - f(A)\psi\|_# \le \epsilon/5$ and $\|g(A_n)\psi - g(A)\psi\|_# \le \epsilon/5$ so by an $\epsilon/3$ argument we get $\|g(A_n)\psi - g(A_n)\psi\|_# \le 3\epsilon/5$. Since the functional calculus implies $\|f(A)\psi - P_{(a,b)}(A)\psi\|_# \le \|f(A)\psi - g(A)\psi\|_#$ another $\epsilon/3$ argument implies $\|P_{(a,b)}(A)\psi - P_{(a,b)}(A)\psi\|_# \le \epsilon$.

**Remark.** Part (a) of Theorem 5.7.7 says that the spectrum of the #-limiting operator cannot suddenly expand. It can, however, contract rather spectacularly as the following example shows: Let $(A_n)_{n=1}^{*\infty} = (x/n)_{n=1}^{*\infty}$ on $L_2^\#(*\tilde{\mathbb{R}}_c^#)$ then $A_n$ #-converges to the zero operator in the strong resolvent sense. For each $n$, $\sigma(A_n) = *\mathbb{R}_c^\#$, but the spectrum of the #-limiting operator contains only the origin. An easy application of part (a) is the statement that if the $A_n$ are positive and $A_n \to_* A$ in the strong resolvent sense, then $A$ is positive.

If $A_n$ #-converges to $A$ in #-norm resolvent sense, Theorem 5.7.6 tells us that the spectrum of the #-limiting operator cannot suddenly contract in the sense that if $\lambda \in \sigma(A_n)$ for all sufficiently infinitely large $n$, then $\lambda \in \sigma(A)$. Notice that in the example $A_n = x/n$ above, $A_n$ does not #-converge to $A$ in the #-norm resolvent sense. The principle of noncontraction of the spectrum under #-norm resolvent #-convergence remains true even when $A_n$ and $A$ are not self-#-adjoint. But the principle of nonexpansion of the spectrum in the strong resolvent #-limit is not always valid for general not-necessarily-self-#-adjoint operators. In fact, there exists a #-norm #-convergent sequence of uniformly bounded operators $A_n \to_* A$ with $\sigma(A_n)$ the unit circle in $*\mathbb{C}_c^\#$ for each $n \in *\mathbb{N}$ and $\sigma(A)$ the entire unit disc. Thus the reader should be careful to apply Theorem 4.7.7 only in the self-#-adjoint case.

In applications, one is usually given the operators $(A_n)_{n=1}^{*\infty}$ and $A$ on domains of
polynomial and $n \in \mathbb{N}^\#$ is arbitrary, is #-dense in $\mathcal{L}_2^\#(Q^\#, d^\#\mu^\#)$. Let $P$ be a polynomial in $N \in \mathbb{N}^\#$ variables

$$P(x_{k_1}, \dots, x_{k_N}) = \mathrm{Ext} - \sum_{l_1, \dots, l_N} c_{l_1, \dots, l_N} x_{k_1}^{l_1} \dots x_{k_N}^{l_N} \quad (8)$$

and define

$$S : P(\varphi^{\#}(f_{k_1}), \dots, \varphi^{\#}(f_{k_N})) \mathbf{Q}_0 \to P(q_{k_1}, \dots, q_{k_N}). \quad (9)$$

Then

$$P(\varphi^{\#}(f_{k_1}), \dots, \varphi^{\#}(f_{k_N})) \mathbf{Q}_0 = \mathrm{Ext} - \sum_{l, \mathbf{m}} c_l \bar{c}_m(\mathbf{Q}_0, \varphi^{\#}(f_{k_1})^{l_1+m_1}, \dots, \varphi^{\#}(f_{k_N})^{l_N+m_N} \mathbf{Q}_0) = \\ \mathrm{Ext} - \sum_{l, \mathbf{m}} c_l \bar{c}_m \int_{\mathbb{R}_c^{\#N}} q_{k_1}^{l_1+m_1} \dots q_{k_N}^{l_N+m_N} \left( \mathrm{Ext} - \prod_{i=1}^N d^\# \mu_{k_i}^\# \right) = \int_{\mathbb{Q}^\#} |P(x_{k_1}, \dots, x_{k_N})|^2 d^\# \mu^\# \quad (10)$$

by (X.92) and the fact that each $\mu_k^\#$ has mass one. Since $\mathbf{Q}_0$ is cyclic for polynomials in the fields (Theorem X.42), $S$ extends to a unitary map of $\mathcal{F}_s^\#(\mathbf{H}^\#)$ onto $\mathcal{L}_2^\#(Q^\#, d^\#\mu^\#)$. Clearly

$$S\varphi^{\#}(f_k)S^{-1} = q_k \quad \text{and} \quad SQ_0 = 1. \quad (11)$$

**Theorem 1.** Let $\varphi_{m,x}^{\#}(f), x \in \mathbb{R}_c^{\#} \mathbb{R}_{c,\mathrm{fin}}^{\#}$ be the free scalar field of mass $m$ (in 4-dimensional space-time) at time zero. Let $g \in \mathcal{L}_1^{\#}(\mathbb{R}_c^{\#3}) \cap \mathcal{L}_2^{\#}(\mathbb{R}_c^{\#3})$ and define

$$H_{l,x,\lambda}(g) = \lambda(x) \int g(x): \varphi_{m,x}^{\#}(x)^4 : d^{\#3}x, \quad (12)$$

where $\lambda(x) \in \mathbb{R}_c^{\#}$, $\lambda(x) \approx 0$. Let $S$ denote the unitary map of $\mathcal{F}_s^\#(\mathbf{H}^\#)$ onto $\mathcal{L}_2^\#(Q^\#, d^\#\mu^\#)$ constructed above. Then $V = SH_{l,x,\lambda}(g)S^{-1}$ is multiplication by a function $V_{x,\lambda}(q)$ which satisfies:

(a) $V_{x,\lambda}(q) \in \mathcal{L}_p^\#(Q^\#, d^\#\mu^\#)$ for all $p \in \mathbb{N}^\#$.

(b) $\mathrm{Ext}\-\exp(-tV_{x,\lambda}(q)) \in \mathcal{L}_1^\#(Q^\#, d^\#\mu^\#)$ for all $t \in [0, \infty)$.

**Proof.** We will prove (a). By Eq.() we get

$$\varphi_{m,x}^{\#}(x) = \frac{1}{(2\pi\#)^{3/2}} \int_{|p| \le x} \{[\mathrm{Ext}\-\exp(-ipx)]a^{\dagger}(p) + [\mathrm{Ext}\-\exp(ipx)]a(p)\} \frac{d^3p}{\sqrt{2\mu(p)}}. \quad (13)$$

Then $\varphi_{m,x}^{\#}(x)$ is a well-defined operator-valued function of $x \in \mathbb{R}_c^{\#3}$. We define $: \varphi_{m,x}^{\#}(x)^4 :$ by moving all the $a^{\dagger}$'s to the left in the formal expression for $\varphi_{m,x}^{\#}(x)^4$.

By Theorem X.44: $\varphi_{m,x}^{\#}(x)^4$: is also a well-defined operator for each $x \in \mathbb{R}_c^{\#3}$ and $: \varphi_{m,x}^{\#}(x)^4 :$ takes $F_0$ into itself. Thus for each $x \in \mathbb{R}_c^{\#3}$,

$$: \varphi_{m,x}^{\#}(x)^4 : = \varphi_{m,x}^{\#}(x)^4 + d_2(x)\varphi_{m,x}^{\#}(x)^2 + d_0(x) \quad (14)$$

where the coefficients $d_2(x)$ and $d_0(x)$ are independent of $x$. For each $x \in \mathbb{R}_c^{\#3}$, $S\varphi_{m,x}^{\#}(x)S^{-1}$ is just the operator on #-measurable space $\mathcal{L}_2^\#(Q^\#, d^\#\mu^\#)$ which operates by multiplying by the function

$$\mathrm{Ext} - \sum_{k=1}^{\infty} c_k(x, x) q_k \quad (15)$$

where
self-#-adjointness or essential self-#-adjointness and it may be very difficult to compute the resolvents. Thus, in order to use Theorem 5.7.6 and Theorem 5.7.7 one must have criteria on the operators $(A_n)_{n=1}^\infty$ and A themselves which guarantee #-norm or strong resolvent #-convergence.

**Theorem 4.5.8.** (a) Let $(A_n)_{n=1}^\infty$ and A be self-#-adjoint operators and suppose that D is a common #-core for all $A_n$ and A. If $A_n\varphi \to_\# A\varphi$ for each $\varphi \in D$ then $A_n \to_\# A$ as $n \to *\infty$ in the strong resolvent sense.

(b) Let $(A_n)_{n=1}^\infty$ and A be self-#-adjoint operators with a common domain, D. Norm D with $||\varphi||_{\#A} = ||A\varphi||_\# \div ||\varphi||_\#$. If $\sup_{||\varphi||_{\#A}=1} (||(A_n - A)\varphi||_\#) \to_\# 0$ as $n \to *\infty$ then $A_n \to_\# A$ in the #-norm resolvent sense.

(c) Let $(A_n)_{n=1}^\infty$ and A be positive self-#-adjoint operators with a common form domain $H_{\pm 1}^\#$, which we norm with $||\psi||_{\#+1} = \langle\psi, A\psi\rangle_\# + \langle\psi, \psi\rangle_\#$. If $A_n \to_\# A$ in #-norm in the sense of maps from $H_{\pm 1}^\#$ to $H_{\pm 1}^\#$ that is, if

$$ \sup_{\psi \neq 0, \varphi \in D} \frac{|\langle \varphi, (A - A_n)\psi \rangle_\#|}{||\varphi||_{\#+1} ||\psi||_{\#+1}} = \sup_{\psi \neq 0, \varphi \in D} \frac{|\langle \varphi, (A - A_n)\psi \rangle_\#|}{\langle \psi, (A+I)\psi \rangle_\#} \to_\# 0 \quad (5.7.18) $$

then $A_n \to_\# A$ in the #-norm resolvent sense.

**Proof** (a) Let $\varphi \in D$, $\psi = (A+i)\varphi$, then $[(A_n + i)^{-1} - (A+i)^{-1}]\psi = (A_n + i)^{-1}(A - A_n)\varphi$ #-converges to zero as $n \to *\infty$, since $(A - A_n)\varphi \to_\# 0$ and the $(A_n + i)^{-1}$ are uniformly bounded. Since D is a #-core for A the set of such $\psi$ is #-dense so for all $\varphi \in H^\# : (A_n + i)^{-1}\varphi \to_\# (A+i)^{-1}\varphi$. A similar proof holds for $(A_n - i)^{-1}$.

We sketch the proofs of (b) and (c). For (b), first one proves that the hypothesis is equivalent to $(A_n - A)(A + i)^{-1} \to_\# 0$ in the ordinary $H^\#$-operator #-norm. Thus $(I + (A_n - A)(A + i)^{-1})^{-1}$ exists and #-converges to I in #-norm as $n \to *\infty$. As a result $(A_n + i)^{-1} = (A + i)^{-1}(I + (A_n - A)(A + i)^{-1})^{-1} \to_\# (A + i)^{-1}$ in #-norm. Similarity $(A_n - i)^{-1} \to_\# (A - i)^{-1}$. To prove (c), we first prove that the hypothesis is equivalent to $(A + I)^{-1/2}(A_n - A)(A + I)^{-1/2} \to_\# 0$ in the ordinary operator #-norm. Using the identity $(A_n + I)^{-1} = (A + I)^{-1/2}[I + (A + I)^{-1/2}(A_n - A)(A + I)^{-1/2}]^{-1}(A + I)^{-1/2}$ one then follows the proof of (b).

## § 5.8. Graph #-limits.

**Definition 5.8.1.** Let $(A_n)_{n=1}^\infty$ be a hyper infinite sequence of operators on a non Archimedean Hilbert space $H^\#$, We say that a pair $\langle\psi, \varphi\rangle_\# \in H^\# \times H^\#$ is in the strong graph #-limit of $A_n$ as $n \to *\infty$ if we can find $\psi_n \in D(A_n)$ so that $\psi_n \to_\# \psi$, $A_n\psi_n \to_\# \varphi$. We denote the set of pairs in the strong graph #-limit by $\Gamma^s_\infty$. If $\Gamma^s_\infty$ is the graph of an operator A we say that A is the strong graph #-limit of $A_n$ and write

$$ A = \text{st. gr.} \text{-#-lim } A_n. \quad (5.8.1) $$

First, we consider the case where all the $A_n$ are self-#-adjoint and A is also self-#-adjoint

**Theorem 5.8.1.** Suppose that $(A_n)_{n=1}^\infty$ and A are self-#-adjoint operators. Then $A_n \to_\# A$ in the strong resolvent sense if and only if $A = \text{st. gr.} \text{-#-lim } A_n$.

**Proof** Suppose first that $(A_n + i)^{-1} \to_\# (A + i)^{-1}$ strongly. Suppose $\varphi \in D(A)$. Then $\varphi_n = (A_n + i)^{-1}(A + i)\varphi \to_\# \varphi$ and $A_n\varphi_n = (A + i)\varphi - i\varphi$, so $\langle\varphi, A\varphi\rangle_\# \in \Gamma^s_\infty$. Thus $\Gamma(A) \subset \Gamma^s_\infty$. On the other hand, suppose $\varphi_n \in D(A_n)$, $\varphi_n \to_\# \varphi$ and $A_n\varphi_n \to_\# \psi$. We let $\eta_n = (A + i)^{-1}(A_n + i)\varphi_n \in D(A)$, then
$$c_k(x, x) = (2\pi\#)^{-3/2} (f_k, \text{Ext}-\exp(ipx)(\mu(p))^{-1/2}). \quad (16)$$

Furthermore,

$$\mathrm{Ext}-\sum_{k=1}^{\infty} |c_k(x, x)|^2 = (2\pi\#)^{-3/2} \|(\mu(p))^{-1/2}\|_2^2, \quad (17)$$

so $S\phi_{m,x}^{\#}(x)^4 S^{-1}$ and $S\phi_{m,x}^{\#}(x)^2 S^{-1}$ are in $\mathcal{L}_2^{\#}(Q^{\#}, d^{\#}\mu^{\#})$ and the $\mathcal{L}_2^{\#}(Q^{\#}, d^{\#}\mu^{\#})$ norms are uniformly bounded in $x$. Therefore, since $g \in \mathcal{L}_1^{\#}(\mathbb{R}_c^{\#3})$, $SH_{I,x,\lambda}(g)S^{-1}$ operates on $\mathcal{L}_2^{\#}(Q^{\#}, d^{\#}\mu^{\#})$ by multiplication by an $\mathcal{L}_2^{\#}(Q^{\#}, d^{\#}\mu^{\#})$ function which we denote by $V_{x,\lambda}(q)$. Consider now the expression for $H_{I,x}(g)Q_0$. This is a vector $(0,0,0,0,\psi^{(4)},0,\dots)$

$$\begin{aligned} \psi^{(4)}(p_1, p_2, p_3, p_4) &= \mathrm{Ext}-\int_{\mathbb{R}_c^{\#3}} \frac{\lambda g(x) [\mathrm{Ext}-\exp(-ix \sum_{i=1}^4 p_i)] d^{\#3}x}{(2\pi\#)^{3/2} \prod_{i=1}^4 (2\mu(p_i))^{1/2}} = \\ &= \frac{\lambda \hat{g}(\sum_{i=1}^4 k_i)}{(2\pi\#)^{9/2} \prod_{i=1}^4 (2\mu(p_i))^{1/2}} \end{aligned} \quad (18)$$

where $|p_i| \le x, 1 \le i \le 4$. We choose now the parameter $\lambda = \lambda(x) \approx 0$ such that $\|\psi^{(4)}\|_2 \in \mathbb{R}$, thus

$$\|H_{I,x,\lambda(x)}(g)Q_0\|_2 \in \mathbb{R}, \quad (19)$$

since $\|H_{I,x,\lambda(x)}(g)Q_0\|_2 = \|\psi^{(4)}\|_2$. But, since $SQ_0 = 1$, we get

$$\|H_{I,x,\lambda(x)}(g)Q_0\|_2 = \|SH_{I,x,\lambda(x)}(g)S^{-1}\|_{\mathcal{L}_2^{\#}(Q^{\#},d^{\#}\mu^{\#})} = \|V_{x,\lambda(x)}(q)\|_{\mathcal{L}_2^{\#}(Q^{\#},d^{\#}\mu^{\#})} \quad (20)$$

From (19) and Eq.(20) we get that $\|V_{x,\lambda(x)}(q)\|_{\mathcal{L}_2^{\#}(Q^{\#},d^{\#}\mu^{\#})}$ is finite. It is easily verify that each $P(q_1,q_2,\dots,q_n), n \in \mathbb{N}^{\#}$ is in the domain of $V_{x,\lambda(x)}(q)$ and $SH_{I,x,\lambda(x)}(g)S^{-1} = V_{x,\lambda(x)}(q)$ on that domain. Since $Q_0$ is in the domain of $[H_{I,x,\lambda(x)}(g)]^p$ for all $n \in \mathbb{N}^{\#}$, $1$ is in the domain of $[V_{x,\lambda(x)}(q)]^n$ for all $n \in \mathbb{N}^{\#}$. Thus, for all $n \in \mathbb{N}^{\#}$, $V_{x,\lambda(x)} \in \mathcal{L}_{2n}^{\#}(Q^{\#},d^{\#}\mu^{\#})$. Since $\mu^{\#}(Q^{\#}) < \infty^{\#}$, $V_{x,\lambda(x)} \in \mathcal{L}_p^{\#}(Q^{\#},d^{\#}\mu^{\#})$ for all $p < \infty^{\#}$.

# Chapter X. A non-Archemedean Banach algebras and $C_{\#}^*$-Algebras.

## §1. A non-Archemedean Banach algebra $B(H^{\#})$

### §1.1. Basic Properties

**Definition 1.1.** An linear operator $T$ on a non-Archemedean Hilbert space $H^{\#}$ is a linear map $H^{\#} \to H^{\#}$. We can define a $\#$-norm by

$$\|T\|_{\#} \triangleq \sup_{v \in H^{\#} \setminus \{0\}} \frac{\|Tv\|_{\#}}{\|v\|_{\#}} \quad (1.1)$$

if supremum in RHS of (1.1) exists.

This is a $\#$-norm since

1. By definition of the $\#$-norm on $H^{\#}$, it is always positive.

2. We have that $T = 0 \Leftrightarrow \forall v \in H^{\#}, Tv = 0 \Leftrightarrow \forall v \in H^{\#} \setminus \{0\}$, $$ \frac{\|Tv\|_{\#}}{\|v\|_{\#}} = 0 \Leftrightarrow \|T\|_{\#} = 0. $$
$$
\begin{align}
\eta_n - \varphi_n &= [(A+i)^{-1} - (A_n+i)^{-1}][(A_n+i)\varphi_n] = \nonumber \\
&= [(A+i)^{-1} - (A_n+i)^{-1}][(A_n+i)\varphi_n - \psi - i\varphi] + \tag{5.8.2} \\
&\quad +[(A+i)^{-1} - (A_n+i)^{-1}][\psi + i\varphi] \xrightarrow{\#} 0 \nonumber
\end{align}
$$

as $n \to *\infty$. Thus, $\eta_n \to_* \varphi$ and $A\eta_n = (A_n+i)\varphi_n - i\eta_n \to_* \psi$ so since A is #-closed
$\langle\varphi, \psi\rangle_* \in \Gamma(A)$. Thus, $\Gamma(A) = \Gamma_*^{s,\infty}$.

Conversely, suppose that $A = \mathbf{st. gr.}$-$\#\text{-lim} A_n$. Let $\varphi \in D(A)$. Then there exist
$\varphi_n \in D(A_n)$ so that $\varphi_n \to_* \varphi$ and $A_n\varphi_n \to_* A\varphi$ as $n \to *\infty$. Thus,
$[(A_n+i)^{-1} - (A+i)^{-1}][(A+i)\varphi] = (A_n+i)^{-1}[(A_n+i)\varphi - (A_n+i)\varphi_n] - (\varphi - \varphi_n) \to_* 0$
as $n \to *\infty$since $\|(A_n+i)^{-1}\|_* \le 1$, $(A_n+i)\varphi_n \to_* (A+i)\varphi$, and $\varphi_n \to_* \varphi$. Since
$\text{Ran}(A+i) = H.^*$ the strong #-convergence of $(A_n+i)^{-1}$ to $(A+i)^{-1}$ follows.

**Remark 5.8.1.** Thus, we see that if the #-limit is self-#-adjoint, then strong graph and strong resolvent #-convergence are the same. It is in the case when we do not know a priori that the #-limit is self-#-adjoint that strong graph #-limits are particularly important. For example, the existence of graph #-limits can sometimes be combined with other information to prove that the #-limit is self-#-adjoint.

**Theorem 5.8.2.** Let $(A_n)_{n=1}^{*\infty}$ be a hyper infinite sequence of symmetric operators.

(a) Let $D_{\infty}^{s} = \{\psi | \langle \psi, \varphi \rangle_{\#} \in \Gamma_{\infty}^{s} \text{ for some } \varphi\}$. If $D_{\infty}^{s}$ is #-dense, then $\Gamma_{\infty}^{s}$ is the graph of an operator.

(b) Suppose that $D_{\infty}^{s}$ is #-dense and let $A = \mathbf{st. gr.}$-$\#\text{-lim} A_n$.
Then A is symmetric and #-closed.

**Proof** We will prove (a); the proof of (b) is obvious. Suppose $\varphi_n, \varphi'_n \in D(A_n)$ and
$\varphi_n \to_* \varphi, \varphi'_n \to_* \varphi$ and $A_n\varphi_n \to_* \psi, A_n\varphi'_n \to_* \psi'$. Let $\eta \in D_{\infty}^s$. Then there is an
$\eta_n \in D(A_n)$, so that $\eta_n \to_* \eta$ and $A_n\eta_n \to_* \rho$ as $n \to *\infty$. Thus,
$\langle\psi - \psi, \eta\rangle_* = \#\text{-lim}_{n \to *\infty} \langle A_n(\varphi_n - \varphi'_n), \eta_n\rangle_* = \#\text{-lim}_{n \to *\infty} \langle\varphi_n - \varphi'_n, A_n\eta_n\rangle_* = 0$
so $\psi = \psi'$ since $D_{\infty}^s$ is #-dense.

We also define weak graph #-limits. We give the definition and state one theorem.

**Definition 5.8.2.** Let $(A_n)_{n=1}^{*\infty}$ be a hyper infinite sequence of operators on $H^\#$. We say that $\langle\psi, \varphi\rangle_* \in H^\# \times H^\#$ is in the weak graph #-limit $\Gamma_*^w$ if we can find $\psi_n \in D(A_n)$ so that $\psi_n \xrightarrow{\|\cdot\|_*} \psi$ and $A_n\psi_n \to_* \varphi$ weakly. If $\Gamma_*^w$ is the graph of an operator, A we say that A is the weak graph #-limit of $A_n$ and abbreviated as $A = \mathbf{w. gr.}$-$\#\text{-lim} A_n$.

**Theorem 5.8.3.** Let $(A_n)_{n=1}^{*\infty}$ be a hyper infinite sequence of symmetric operators. If $D_{\infty}^{s} = \{\psi | \langle \psi, \varphi \rangle_{\#} \in \Gamma_{\infty}^{s} \text{ for some } \varphi\}$ is #-dense, then $\Gamma_{\infty}^{w}$ is the graph of a symmetric operator.

**Remark 5.8.2.** Finally we note that if $A_n$ is a uniformly bounded sequence of operators then $A = \mathbf{w. gr.}$-$\#\text{-lim} A_n$ if and only if $A_n \xrightarrow{\#} A$ as $n \to *\infty$ in the weak operator topology. This fact shows that the notions of weak graph #-limit and weak resolvent #-convergence are distinct. It is not true that weak graph #-limits are necessarily #-closed if each $A_n$ is symmetric.

§ 5.9. Generalized Trotter product formula

**Theorem 4.9.1.** (Generalized Lie product formula) Let A and B be external hyperfinite-dimensional matrices. Then

$$
\operatorname{Ext}\nolimits -\operatorname{exp}(A + B) = \#\operatorname{lim}_{n \to *\infty} \{[\operatorname{Ext}\nolimits -\operatorname{exp}(A/n)] \times [\operatorname{Ext}\nolimits -\operatorname{exp}(B/n)]\}^n. \quad (5.9.1)
$$
$$3. \|\lambda T\|_{\#} = \sup_{v \in H^{\#} \setminus \{0\}} \frac{\|\lambda T v\|_{\#}}{\|v\|_{\#}} = |\lambda| \sup_{v \in H^{\#} \setminus \{0\}} \frac{\|T v\|_{\#}}{\|v\|_{\#}} = |\lambda| \|T\|$$

$$4. \|T_1 + T_2\|_{\#} = \sup_{v \in H^{\#} \setminus \{0\}} \frac{\|T_1 v + T_2 v\|_{\#}}{\|v\|_{\#}} \le \sup_{v \in H^{\#} \setminus \{0\}} \frac{\|T_1 v\|_{\#} + \|T_2 v\|_{\#}}{\|v\|_{\#}} \le \\ \le \sup_{v \in H^{\#} \setminus \{0\}} \frac{\|T_1 v\|_{\#}}{\|v\|_{\#}} + \sup_{v \in H^{\#} \setminus \{0\}} \frac{\|T_2 v\|_{\#}}{\|v\|_{\#}} = \|T_1\|_{\#} + \|T_2\|_{\#}.$$

**Definition 1.2.** Let $H^\#$ be a non-Archemedean Hilbert space over $\mathbb{C}_c^\#$. A linear map $A: H^\# \to H^\#$ is called bounded in $\mathbb{R}_c^\#$ operator iff $\|A\|_\# < *\infty$.

**Definition 1.3.** Let $H^\#$ be a non-Archemedean Hilbert space over $\mathbb{C}_c^\#$. We denote by $B(H^\#)$ the set of all bounded in $\mathbb{R}_c^\#$ operators $A: H^\# \to H^\#$.

**Definition 1.4.** Algebra $A$ is called an algebra over $\mathbb{C}_c^\#$ if it is a vector space over $\mathbb{C}_c^\#$ and a binary map $\cdot: A \times A \to A$ Satisfying:

1. Left distributivity: $\forall v, w, u \in A[(v + w) \cdot u = v \cdot u + w \cdot w]$

2. Right distributivity: $\forall v, w, u \in A[v \cdot (w + u) = v \cdot w + v \cdot u]$

3. $\forall v, w \in A, \forall \alpha, \beta \in \mathbb{C}_c^\# [\alpha\beta v \cdot w = (\alpha v) \cdot (\beta w)]$

We note that $B(H^\#)$ is an algebra over $\mathbb{C}_c^\#$ where for $A, B \in B(H^\#)$, $\lambda \in \mathbb{C}_c^\#$ we define:

$$\lambda A: H^{\#} \to H^{\#}, \nu \mapsto \lambda A \nu$$

$$A + B: H^{\#} \to H^{\#}, \nu \mapsto A \nu + B \nu$$

$$A \cdot B: H^{\#} \to H^{\#}, \nu \mapsto A(B(v))$$

In $B(H^\#)$ we have the #-adjoint operator. This maps each $A$ to the unique $A^*$ such that for all $v, w \in H^\#$ we have $\langle Av, w \rangle_\# = \langle v, A^*w \rangle_\#$. We denote the adjoint of an operator $A$ by $A^*$ and define the adjoint of a subset $M \subset B(H^\#)$ by

$$M^* \triangleq \{A^* \in B(H^\#) | A \in M\}.$$ The adjoint has the following key properties:

**Lemma 1.4.** Adjoint Properties (Algebraic)

For $A, B \in B(H^\#)$ we have

1. $A^*$ always exists is unique.

2. If $A$ is bounded in $\mathbb{R}_c^\#$, then $A^*$ is also bounded in $\mathbb{R}_c^\#$.

3. $A^{**} = A$ (Involutivity)

$$4. \|A\|_{\#} = \|A^*\|_{\#}$$

5. If $A$ is invertible, $A^*$ also is, with $(A^*)^{-1} = (A^{-1})^*$

$$6. (A + B)^* = A^* + B^*, (\lambda A)^* = \bar{\lambda} A^*$$

$$7. (AB)^* = B^* A^*$$

$$8. \|A^*A\|_{\#} = \|A\|_{\#}$$

**Proof.** 1. Let $x \in H^\#$ and consider the bounded in $\mathbb{R}_c^\#$ linear functional $f: H^\# \to \mathbb{C}_c^\#$, $f(v) \mapsto \langle Av, x \rangle_\#$ we have $\|f\| \le \|A\|_\# \|x\|_\#$. By generalized Riesz representation theorem there exists a unique $y \in H^\#$ with $f(v) = \langle v, y \rangle_\# \forall v \in H^\#$. So we set $A^*x = y$. Then for any $y, z \in H^\#$ and $\forall \alpha \in \mathbb{C}_c^\#$ we have:

$$\langle v, A^*(\alpha y + z) \rangle_\# = \langle Av, \alpha y + z \rangle_\# = \bar{\alpha} \langle Av, y \rangle_\# + \langle Av, z \rangle_\# = \bar{\alpha} (\langle v, A^*y \rangle_\# + \langle v, A^*z \rangle_\#) = \\ = \langle v, \alpha A^*y + A^*z \rangle_\# \quad \forall v \in H^\#. \text{ In particular, if we choose } v = A^*(\alpha y + z) - \alpha A^*y + A^*z, \\ \text{we see that } \|v\|_\# = 0 \Rightarrow v = 0 \Rightarrow A^* \text{ is linear.}$$

2. Following from 1. we have

$$\|A^*x\|_{\#} = \|y\|_{\#} = \|f\|_{\#} \le \|A\|_{\#} \|y\|_{\#}.$$

3. We can see this as

$$\langle A^{**}v, w \rangle_{\#} = \langle v, A^{*}w \rangle_{\#} = \langle Av, w \rangle_{\#} \quad \forall v, w \in H^{\#}.$$
**Proof** Let $S_n = \text{Ext-exp}((A+B)/n)$ and $T_n = [\text{Ext-exp}(A/n)] \times [\text{Ext-exp}(B/n)]$. Then

$$S_n^n - T_n^n = \text{Ext-}\sum_{m=0}^{n-1} S_m^n(S_n - T_n)T_n^{n-m-1} \quad (5.9.2)$$

so that

$$\begin{aligned} \|S_n^n - T_n^n\|_{\#} &\le n(\max(\|S_n\|_{\#}, \|T_n\|_{\#}))^{n-1} \|S_n - T_n\|_{\#} \\ &\le n \|S_n - T_n\|_{\#} [\text{Ext-exp}(\|A\|_{\#} + \|B\|_{\#})]. \end{aligned} \quad (5.9.3)$$

Since

$$\begin{aligned} \|S_n - T_n\|_{\#} &= \left\| \text{Ext-}\sum_{m=0}^{\infty} \frac{1}{m!^{\#}} \left(\frac{A+B}{n}\right)^m - \left( \text{Ext-}\sum_{m=0}^{\infty} \frac{1}{m!^{\#}} \left(\frac{A}{n}\right)^m \right) \left( \text{Ext-}\sum_{m=0}^{\infty} \frac{1}{m!^{\#}} \left(\frac{B}{n}\right)^m \right) \right\|_{\#} \\ &\le C/n \end{aligned} \quad (5.9.4)$$

where constant $C$ depends only on $\|A\|_{\#}$ and $\|B\|_{\#}$ we conclude that $\#\lim_{n \to \infty} \|S_n^n - T_n^n\|_{\#} = 0$.

**Remark 5.9.1.** This theorem and its proof can be extended to the case where $A$ and $B$ are unbounded self-#-adjoint operators and $A+B$ is self-#-adjoint on $D(A) \cap D(B)$.

**Theorem 5.9.2** Let $A$ and $B$ be self-#-adjoint operators on $H^\#$ and suppose that $A+B$ is self-#-adjoint on $D = D(A) \cap D(B)$. Then

$$\text{Ext-exp}[it(A+B)] = \mathbf{s}\text{-}\lim_{n \to \infty} \{[\text{Ext-exp}(itA/n)] \times [\text{Ext-exp}(itB/n)]\}^n. \quad (5.9.5)$$

**Proof** Let $\psi \in D$. Then

$$\begin{aligned} s^{-1}\{[\text{Ext-exp}(isA)] \times [\text{Ext-exp}(isB)] - I\}\psi &= s^{-1}\{[\text{Ext-exp}(isA)] - I\}\psi + s^{-1}\{[\text{Ext-exp}(isB)] - I\}\psi \to_{\#} i(A+B)\psi \end{aligned} \quad (5.9.6)$$

and

$$s^{-1}\{[\text{Ext-exp}(isA)] \times [\text{Ext-exp}(isB)] - I\}\psi \to_{\#} i(A+B)\psi \quad (5.9.7)$$

as $s \to \# 0$. Letting

$$K(s) = s^{-1}\{[\text{Ext-exp}(isA)] \times [\text{Ext-exp}(isB)] - [\text{Ext-exp}(is(A+B))]\} \quad (5.9.8)$$

we see that $K(s)\psi \to_{\#} 0$ as $s \to_{\#} 0$, for each $\psi \in D$. Since $A+B$ is self-#-adjoint on $D$, $D$ is a

Banach space under the #-norm

$$\|\psi\|_{\#A+B} = \|(A+B)\psi\|_{\#} + \|\psi\|_{\#}. \quad (5.9.9)$$

Each of the maps $K(s) : D \to H^\#$ is bounded and $K(s)\psi \xrightarrow{s \to \#} 0$ as $s \to \# 0$ or $\infty$ for each $\psi \in D$.

Thus, we conclude from the uniform boundedness theorem that the $K(s)$ are uniformly bounded, that is, there is a constant $C$ so that $\|K(s)\psi\|_{\#} \le C\|\psi\|_{\#A+B}$ for all $s \in \mathbb{R}_c^\#$ and $\psi \in D$. Therefore, an $\varepsilon/3$ argument shows that on $\|\cdot\|_{\#A+B}$ #-compact subsets of $D$ $K(s)\psi \to_{\#} 0$ uniformly. Since $A+B$ is self-#-adjoint on $D$, $[\text{Ext-exp}(is(A+B))]\psi \in D$ if $\psi \in D$. Moreover, $s \to [\text{Ext-exp}(is(A+B))]\psi$ is a #-continuous map of $\mathbb{R}_c^\#$ into $D$ when $D$ is given the $\|\cdot\|_{\#A+B}$ #-norm topology. Thus $\{[\text{Ext-exp}(is(A+B))]\psi|s \in [-1, 1]\}$ is a $\|\cdot\|_{\#A+B}$ #-compact set in $D$ for each fixed $\psi$.
4. Combining the estimate from above and involutivity, we have
$$ ||A^{**}||_{#} \le ||A^{*}||_{#} \le ||A||_{#} = ||A^{**}||_{#}. $$

So we must have equality everywhere.

5. We have $\langle v, (A^{-1})^*A^*w \rangle_# = \langle A^{-1}v, A^*w \rangle_# = \langle AA^{-1}v, w \rangle_# = \langle v, w \rangle_# \quad \forall v, w \in H^#.$

Hence, $(A^{-1})^*A^* = 1$. The argument for $A^*(A^{-1})^* = 1$ is the same.

6. This follows clearly from conjugate linearity in the second argument of an inner product.

7. This is clear since, $\langle ABv, w \rangle_# = \langle Bv, A^*w \rangle_# = \langle v, B^*A^*w \rangle_# \quad \forall v, w \in H^#.$

8. For this we have $||T||_#^2 = \sup_{||x||_# = 1} ||Tx||_#^2 = \sup_{||x||_# = 1} |\langle Tx, Tx \rangle_#| = \sup_{||x||_# = 1} |\langle T^*Tx, x \rangle_#| \le \sup_{||x||_# = 1} ||T^*Tx||_# ||x||_# = ||T^*T||_#$. But also, $||T^*T||_# \le ||T^*||_# ||T||_# = ||T||_#^2$, and so there is equality everywhere.

## §1.2 Types of Operators

**Definition 1.2.1.** A is called normal if $A^*A = AA^*$.

**Definition 1.2.2.** A is called positive if $A = B^*B$ for some $B \in B(H^#)$

**Definition 1.2.3.** A is called self #-adjoint if $A^* = A$.

**Lemma 1.2.1.** Let $A \in B(H^#)$. Then $A = A_1 + iA_2$ where $A_1$ and $A_2$ are both self #-adjoint.

**Proof.** Let $A_1 = \frac{A+A^*}{2}$, $A_2 = \frac{iA^*-iA}{2}$.

It is then clear from basic algebra.

**Definition 1.2.4.** U is called unitary if $U^*U = UU^* = 1$

**Example 1.2.1.** If U is unitary, we have $\forall h, k \in H^#$, $\langle h, k \rangle_# = \langle Uh, Uk \rangle_#$. This is because $\langle Uh, Uk \rangle_# = \langle h, Uk \rangle_# = \langle h, 1k \rangle_# = \langle h, k \rangle_#$.

**Definition 1.2.5.** A is called isometric if $A^*A = 1$.

We also have a relaxed definition, a partial isometry.

**Definition 1.2.6.** A is called a partial isometry if it is an isometry on the orthogonal complement of its kernel, i.e. $A^*A v = v, \forall v \in \text{Ker}(A)^\perp =$ $= \{v \in H^# | \langle v, w \rangle_# = 0, \forall w \in \text{Ker}(A)\}$.

**Definition 1.2.7.** $p \in B(H^#)$ is called a projection if $p = p^* = p^2$.

**Example 1.2.2.** Consider $H^# = l_2^#(*\mathbb{N})$ the set of all square summable $*C_c^#$-valued series. An example of a projection would be:

$p_n:H^# \to H^#$, $(a_1, a_2, \dots, a_n, a_{n+1}, a_{n+2}, \dots) \mapsto (a_1, a_2, \dots, a_n, 0, 0, \dots)$.

We see this is self #-adjoint as $\langle p_n a, b \rangle_# = \text{Ext}-\sum_{k=1}^n a_k \bar{b}_k = \langle a, p_n b \rangle_#$ and idempotent as $p_n^2 = p_n$.

**Lemma 1.2.2.** Multiplication and #-norm property

$\forall A, B \in B(H^#), ||A \cdot B||_# \le ||A||_# ||B||_#$

**Proof.** For all $h \in H^#$, we always have the estimate $||Ah||_# \le ||A||_# ||h||_#$.

Using this we have

$$ ||AB||_# = \sup_{h \in H \setminus \langle 0 \rangle} ||(AB)h||_# / ||h||_# = \sup_{h \in H \setminus \langle 0 \rangle} ||A(Bh)||_# / ||h||_# $$

$$ \le \sup_{h \in H \setminus \langle 0 \rangle} ||A||_# ||Bh||_# / ||h||_# = ||A||_# ||B||_# $$

**Lemma 1.2.3.** $(B(H^#), ||\cdot||_#)$ is complete, i.e. if $(A_n)_{n \in *\mathbb{N}} \subset B(H^#)$ is cauchy with respect to the operator #-norm $||\cdot||_#$, it #-converges in #-norm to some element $A \in B(H^#)$.

**Proof.** Let $(A_n)_{n \in *\mathbb{N}}$ be cauchy with respect to the operator #-norm. This means that $\forall \varepsilon(\varepsilon \approx 0, \varepsilon > 0) \exists N \in *\mathbb{N}_\infty [n, m > N \Rightarrow ||A_n - A_m||_# < \varepsilon]$.
We are now ready to mimic the proof of the generalized Lie product formula. We
know that

$$
t^{-1} \{[\mathrm{Ext}-\exp(itA)] \times [\mathrm{Ext}-\exp(itB)] - [\mathrm{Ext}-\exp(it(A+B))]\} \times \\
\times [\mathrm{Ext}-\exp(is(A+B))]\psi \to_{\#} 0 \tag{5.9.10}
$$

uniformly for $s \in [-1, 1]$. Therefore, we write

$$
\begin{align}
& (\{[\mathrm{Ext}-\exp(itA/n)] \times [\mathrm{Ext}-\exp(itB/n)]\}^n - [\mathrm{Ext}-\exp(it(A+B)/n)]^n)\psi = \nonumber \\
& \mathrm{Ext}-\sum_{k=0}^{n} \{[\mathrm{Ext}-\exp(itA/n)] \times [\mathrm{Ext}-\exp(itB/n)]\}^k \times \nonumber \\
& [(\{[\mathrm{Ext}-\exp(itA/n)] \times [\mathrm{Ext}-\exp(itB/n)]\} - [\mathrm{Ext}-\exp(it(A+B)/n)]) \times \nonumber \\
& \qquad [\mathrm{Ext}-\exp(it(A+B)/n)]^{n-k-1}\psi \tag{5.9.11}
\end{align}
$$

The #-norm of the RHS of (4.9.11)

$$
\max_{|s|<t} \left\| \left(\frac{t}{n}\right)^{-1} \left\{ \left[ \left[ \mathrm{Ext}-\exp\left(it(A+B)/n\right) \right] - \left[ \left[ \mathrm{Ext}-\exp\left(itA/n\right) \right] \times \left[ \mathrm{Ext}-\exp\left(itB/n\right) \right] \right] \right\} \right\}_{\#} \quad (4.9.12)
$$

and so we conclude that

$$
\{[\mathrm{Ext}-\exp(itA/n)] \times [\mathrm{Ext}-\exp(itB/n)]\}^n \psi \xrightarrow{H^\#} \mathrm{Ext}-\exp(it(A+B))\psi \quad (4.9.13)
$$

as $n \to *\infty$ if $\psi \in D$; Since $D$ is #-dense and the operators are bounded by one, this statement holds on all of $H^\#$. The above proof shows that on a fixed vector the #-convergence is uniform for $t$ in a #-compact subset of $*\mathbb{R}_c^\#$.

**Remark 5.9.2.** The same argument can be used to show that

$$
\mathbf{s}\text{-}\mathbf{s}\text{-}\lim_{n\to\infty} \{[\mathrm{Ext}\text{-}\exp(itA/n)] \times [\mathrm{Ext}\text{-}\exp(itB/n)]\}^n = \mathrm{Ext}\text{-}\exp(t(A+B)) \quad (5.9.14)
$$

if A and B satisfy the same hypotheses and in addition are semibounded. The
following

result is considerably stronger than Theorem 4.9.2 since it only requires essential
self-#-adjoint ness of $A + B$ on $D(A) \cap D(B)$.

**Theorem 5.9.3** (the generalized Trotter product formula) If $A$ and $B$ are self-#-adjoint operators and $A+B$ is essentially self-adjoint on $D(A) \cap D(B)$ then

$$
\mathbf{s}\text{-}\mathbf{s}\text{-}\lim_{n\to\infty} \{[\mathrm{Ext}\text{-}\exp(itA/n)] \times [\mathrm{Ext}\text{-}\exp(itB/n)]\}^n = \mathrm{Ext}\text{-}\exp(it(A+B)) \quad (5.9.15)
$$

Moreover, if A and B are bounded from below, then

$$
\mathbf{s}\text{-}\mathbf{s}\text{-}\lim_{n\to\infty} \{[\mathrm{Ext}\text{-}\exp(-tA/n)] \times [\mathrm{Ext}\text{-}\exp(-tB/n)]\}^n = \mathrm{Ext}\text{-}\exp(-t(A+B)). \quad (4.9.16)
$$

§ 5.10. The polar decomposition

Note that an arbitrary bounded operator $T$ can be written $T = U|T|$ where $|T|$ is positive
and self-#-adjoint and $U$ is a partial isometry. Moreover, the conditions that $\mathrm{Ker}(|T|) =$
$= \mathrm{Ker}(T)$ and that the initial space of $U$ equals $(\mathrm{Ker}(T))^{\perp}$ uniquely determine $|T|$ and
$U$. In this section we extend this result to closed #-unbounded operators. As in the
bounded case, $U$ is easy to construct once $|T|$ has been constructed and, as in
the bounded case, we will let $|T| = \sqrt{T^*T}$. In the bounded case, the hard part was the
construction of the square root. Now that we have the spectral
theorem, it is easy to construct $\sqrt{T^*T}$ if we can prove that $T^*T$ is a positive self-
In particular then $\|A_n\|$ is bounded above, say by $K \in *\mathbb{R}_{c,+}^\#$. Now fix $v \in H^\#$ and let $N, m, n \in *\mathbb{N}_\infty$ be as before. We have that

$$ \|A_n v - A_m v\|_{\#} \le \|A_n - A_m\|_{\#} \|v\|_{\#} \le \|v\|_{\#}. $$

Hence, $(A_n v)_{n \in *\mathbb{N}}$ is cauchy in $H^\#$. By completeness of $H^\#$, we have a $\#$-limit and can define $A:H^\# \to H^\#$, $v \mapsto \#\text{-lim}_{n \to *\infty} A_n v$, this is our candidate for our $\#$-limit.

A is linear since (by algebra of $\#$-limits)

$$ A(\alpha v + w) = \#\text{-lim}_{n \to *\infty} A_n(\alpha v + w) = \alpha (\#\text{-lim}_{n \to *\infty} A_n v) + \#\text{-lim}_{n \to *\infty} A_n w = \alpha Av + Aw $$
and bounded in $*\mathbb{R}_{c,+}^\#$ because

$$ \|Av\|_{\#} = \#\text{-lim}_{n \to *\infty} \|A_n v\|_{\#} = \#\text{-lim}_{n \to *\infty} \|A_n v\|_{\#} \le \#\text{-lim}_{n \to *\infty} \|A_n\|_{\#} \|v\|_{\#} \le K\|v\|_{\#}. $$

Hence, $A \in B(H^\#)$. Finally we show convergence in $\#$-norm. Fix $\varepsilon \approx 0, \varepsilon > 0$ and let $N \in *\mathbb{N}_\infty$ be as in the definition of cauchy. If $n > N$ we have:

$$ \|A - A_n\|_{\#} = \sup_{\|v\|_{\#} = 1} \|\#\text{-lim}_{m \to *\infty} (A_m - A_n)v\|_{\#} \le $$

$$ \sup_{\|v\|_{\#} = 1} \#\text{-lim}_{m \to *\infty} \|(A_m - A_n)v\|_{\#} \|v\|_{\#} \le \#\text{-lim}_{m \to *\infty} = \varepsilon. $$

**Definition 1.2.8.** $A \in B(H^\#)$ is $\#$-compact if for all bounded subsets $\beta$ of $H^\#$ the image of $A$ restricted to $\beta$ has $\#$-compact $\#$-closure:

$$ A \in B(H^\#) \text{ $\#$-compact} \Leftrightarrow \forall \beta \text{ bounded, } \#\overline{A\beta} \text{ is $\#$-compact.} $$

We denote by $K^\#(B(H^\#))$ the set of all $\#$-compact operators in $B(H^\#)$.

**Lemma 1.2.4.** A is $\#$-compact iff

$$ \forall (v_\alpha)_{\alpha \in X} \text{ bounded} \Rightarrow (Av_\alpha)_{\alpha \in X} \text{ has a } \#$-convergent subsequence. $$

**Definition 1.2.9.** For $A \in B(H^\#)$ the rank $\Re(A)$ of $A$ is the dimension of the range of $A$

**Lemma 1.2.5.** If $A$ has rank $N \in *\mathbb{N}$, then we can write

$$ A(\cdot) = \text{Ext}-\sum_{n \le N} \alpha_n \langle \cdot, v_n \rangle_{\#} w_n \text{ where } \{v_n\}_{n \le N}, \{w_n\}_{n \le N} \subset H^\#, \{\alpha_n\}_{n \le N} \subset *\mathbb{C}_c^\# $$

**Proof.** This follows immediately from the generalized Riesz representation theorem noting that if $\{w_n\}_{n \le N}$ forms a basis of $A(H^\#)$, then $\langle A \cdot, w_n \rangle_{\#}$ is a linear functional $H^\# \to *\mathbb{C}_c^\#$. So for some $v_n$, $\langle A \cdot, w_n \rangle_{\#} = \langle \cdot, v_n \rangle_{\#}$ so setting $\alpha_n = 1/\|w_n\|_{\#}$ we have

$$ A \cdot = \text{Ext}-\sum_n \alpha_n \langle A \cdot, w_n \rangle_{\#} w_n = \text{Ext}-\sum_{n \le N} \alpha_n \langle \cdot, v_n \rangle_{\#} w_n. $$

We denote by $\mathcal{F}^\#\Re(B(H^\#))$ the set of hyperfinite rank operators in $B(H^\#)$.

**Lemma 1.2.6.** Any operator with hyperfinite rank is $\#$-compact

**Proof.** Say $A$ has hyperfinite rank, then if $(v_\alpha)_{\alpha \in X}$ is bounded, then $(Av_\alpha)_{\alpha \in X}$ is bounded, and lies in a hyperfinite dimensional Hilbert space. By generalized Bolzano Weirstrass theorem [1], we have that it omits a $\#$-convergence subsequence so by lemma 1.2.5, $A$ is $\#$-compact.

**Theorem 1.2.1.** Let $H_1 = \{h \in H^\# \|\|h\|_\# < 1\}$ The following are equivalent:

1. $A \in \overline{\{C \mid \Re(C) < *\infty\}} = \overline{\mathcal{F}^\#\Re(B(H^\#))}$ where the $\#$-closure is with respect to the $\#$-norm topology

2. $A \in K^\#(B(H^\#))$

3. $A(H_1)$ has $\#$-compact $\#$-closure

**Proof.** $1 \Rightarrow 2$.

If $A \in \overline{\{C \mid \Re(C) < *\infty\}}$ then the $\#$-compactness $A$ is clear, since hyperfinite rank operators are in $K^\#(B(H^\#))$ and $K^\#(B(H^\#))$ must be $\#$-closed with respect to the $\#$-norm topology.

2 $\Rightarrow$ 3.

This is clear by definition since $H_1$ is bounded subset of $H^\#$.

3 $\Rightarrow$ 1.

Say this did not hold, i.e. we had some $A$ that has property 3 but not 1. Then, let
#-adjoint operator. It is this fact that is hard in the unbounded case. A priori, it is not clear that {$\psi|\psi \in D(T)$ and $T\psi \in D(T^*)$} is different from {$0$}. In fact, this set is #-dense, but our approach using the theory of semi-bounded quadratic forms does not require us to prove this.

**Theorem 5.10.1.** (the polar decomposition) Let $\Gamma$ be an arbitrary #-closed operator on a

non Archimedean Hilbert space $H^#$. Then, there is a positive self-adjoint operator $|T|$, with $D(|T|) = D(T)$ and a partial isometry $U$ with initial space, $(\mathbf{Ker}(T))^\perp$, and final space #-Ran($T$) so that $T = U|T|$ and $U$ are uniquely determined by these properties together with the additional condition $\mathbf{Ker}(|T|) = \mathbf{Ker}(T)$.

**Proof.** Define the *C#-valued quadratic form $s(\psi, \varphi)$ on $D(T)$ by

$$s(\psi, \varphi) = \langle T\psi, T\varphi \rangle_{\#}. \qquad (5.10.1)$$

Quadratic form $s(\psi, \varphi)$ is clearly positive. Now suppose $\|\psi_n - \psi_m\|_{\#+1} \to \# 0$. Then $\|\psi_n - \psi_m\|_{\#} \to \# 0$ and $\|T(\psi_n - \psi_m)\|_{\#} \to \# 0$. Since $T$ is #-closed there is a $\psi \in D(T)$ with

$$\|\psi_n - \psi\|_{\#} + \|T(\psi_n - \psi)\|_{\#} \to \# 0, \qquad (5.10.2)$$

i.e. $\|\psi_n - \psi\|_{\#+1} \to \# 0$.

Thus $s(\psi, \varphi)$ is a #-closed form. Therefore, by **Theorem VIII.15**, there is a unique, positive self-#-adjoint operator $S$ with $Q(S) = D(T)$ and $s(\psi, \varphi) = \langle \psi, S\varphi \rangle_{\#}$ in the sense of *C#-valued quadratic forms*. Let $|T| = S^{1/2}$. Then $D(|T|) = Q(S) = D(T)$ and by construction $\|T|\psi\|_{\#}^2 = s(\psi, \psi) = \|T\psi\|_{\#}^2$ so $\mathbf{Ker}(|T|) = \mathbf{Ker}(T)$. Define the operator $U: \mathbf{Ran}(|T|) \to \mathbf{Ran}(T)$ by $U|T|\psi = T\psi$. Since $\|T|\psi\|_{\#} = \|T\psi\|_{\#}$, $U$ is well defined and #-norm preserving. Thus $U$ extends to a partial isometry from $\mathbf{Ran}(|T|)$ to $\mathbf{Ran}(T)$. Finally, since $|T|$ is self-#-adjoint, #-Ran($T$) = $(\mathbf{Ker}(|T|))^\perp = (\mathbf{Ker}(T))^\perp$. Uniqueness is obvious.

# 6. Tensor products and second quantization.

## 6.1. Tensor products.

In this section we describe some aspects of the theory of tensor products of operators on non Archimedean Hilbert spaces. Let $A$ and $B$ be #-densely defined operators on non Archimedean Hilbert spaces $H_1^\#$ and $H_2^\#$ respectively. We will denote by $D(A) \otimes D(B)$ the set of hyperfinite linear combinations of vectors of the form $\varphi \otimes \psi$ where $\varphi \in D(A)$ and $\psi \in D(B)$. $D(A) \otimes D(B)$ is dense in $H_1^\# \otimes H_2^\#$ We define $A \otimes B$ on $D(A) \otimes D(B)$ by

$$A \otimes B(\varphi \otimes \psi) = A\varphi \otimes B\psi \qquad (6.1.1)$$

and extend by linearity.

**Proposition 5.1.1** The operator $A \otimes B$ is well defined. Further, if $A$ and $B$ are #-closable,

so is $A \otimes B$.

**Proof** Suppose that $\mathrm{Ext}-\sum c_i \varphi_i \otimes \psi_i$ and $\mathrm{Ext}-\sum d_j \varphi'_j \otimes \psi'_j$ are two representations of the same vector $f \in D(A) \otimes D(B)$. Using Gram-Schmidt orthogonalization we obtain bases $\{\eta_k\}$ and $\{\theta_l\}$ for the spaces spanned by $\{\varphi_i\} \cup \{\varphi'_i\}$ and $\{\psi_j\} \cup \{\psi'_j\}$ respectively so that $\eta_k \in D(A)$ and $\theta_l \in D(B)$. $\varphi_i \otimes \psi_i$ and $\varphi'_j \otimes \psi'_j$ can be expressed