数理解析研究所講究録
第 855 巻 1993 年 118-141

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Toward Harmonic Analysis on Gaussian SpaceNOBUAKI OBATA(尾畑伸明)

DEPARTMENT OF MATHEMATICSSCHOOL OF SCIENCENAGOYA UNIVERSITYNAGOYA, 464-01 JAPANIntroduction
As is well known, there are two important aspects of Gaussian space: stochastic
analysis and quantum field theory. Needless to say, in these theories most principal
roles are played respectively by Brownian motion and Fock space both of which arerealized on Gaussian space. Thus it is widely accepted that Gaussian space is one of
the most important concepts of infinite dimensional analysis such as Euclidean space infinite dimensional case. Moreover, since $1970s$ distribution theories on Gaussian spacehave developed considerably into a most flourishing field of mathematics.
On the other hand, it is remarkable that some pioneering works were made by japanesemathematicians in $1960s$ toward ”harmonic analysis on Gaussian space.” A central
object was perhaps the infinite dimensional rotation group $O(E;H)$ proposed by H.
Yoshizawa after his study of unitary representations of free groups. During that decadea series of important works appeared discussing infinite dimensional Laplacian, infinite
dimensional rotation group, infinite dimensional motion group, and special functions
as matrix elements of their unitary representations, see K\^ono [9], [10], Orihara [26],Umemura [27], Umemura and K\^ono [28]. Furthermore, it was shown by Hida, Kubo,
Nomoto and Yoshizawa [5] and Yoshizawa [30] that the infinite dimensional rotationgroup plays also an important role in describing projective invariance of Brownian motion, see also [31]. However, little progress has been made afterward and, in particular,
no special attention has been paid to application of distribution theories born in $1970s$ .
In recent years the so-called white noise calculus, a distribution theory on Gaussian
space initiated by Hida [3] and axiomatized to some extent by Kubo and Takenaka[11], has developed considerably keeping a profound contact with stochastic (causal)
analysis and Feynman path integrals, see e.g., [7]. Meanwhile, establishing a general
theory of operators on white noise functionals using integral kernel operators and Fockexpansion, we have started a study of harmonic analysis on Gaussian space, see also[21]. One of the main consequences of our operator theory is that every continuous

119linear operator on white noise functionals (this class contains all bounded operators on
Fock space) admits an infinite series expansion in terms of creation and annihilation
operators. This theory is highlighted in [22] and [23], see also \S 6.
The main purpose of this paper is to recapitulate the operator theory on Gaussian
space with illustrating application to some questions of harmonic analysis, in particular,
to description of rotation-invariant operators.
In his important work [27] Umemura showed that “rotation-invariant operators” aregenerated by a single operator, namely, by the number operator $N$ . However, we can
not help ourselves feeling that the structure of the rotation-invariant operators is evenpoorer, comparing to the finite dimensional case. Moreover, during the derivation of
the number operator from finite dimensional Laplacians by limit argument, Umemura
abandoned polynomial terms simply by reason of divergence. We shall observe thatwhite noise calculus explains it to some extent. In fact, in our sense the rotationinvariant operators on Gaussian space are gerenated by two Laplacians, the number
operator $N$ and the Gross Laplacian. Note, however, that there is no contradiction
between Umemura’s work and our result. The point is very simple: the Gross Laplacian. Furthermore, a white noise
is not symmetric and the proper -domain ofisanalogue of Euclidean norm is given by $R=2N+\Delta_{G}+\Delta_{G}^{*}$ . We shall observe thatis extracted from the “divergent terms” in Umemura’s argument. Thus, within white
noise calculus the structure of rotation-invariant operators is more similar to the finite
dimensional case.
$\Delta_{G}$$L^{2}$$\Delta_{G}^{*}$$\{0\}$$R$

1. Gaussian SpaceLet be a topological space with a Borel measure $\nu(dt)=dt$ and let $H=L^{2}(T, \nu;R)$v-square integrable functions on . The inner productbe the real Hilbert space ofis denoted by \langle ., ) and the norm by. We often regard as time-parameter spacee.g., when $T=R,$ , and as space-time-parameter space in quantum field theory e.g.,when $T=R^{D},$ .
Let
be a positive selfadjoint operator on $H$ with Hilbert-Schmidt inverse. Then
there exist an increasing sequence of positive numbers
and acomplete orthonormal basis
for $H$ such that
and
$T$

au

$T$

$|\cdot|_{0}$$\cdot$$T$$Z$$Z^{D}$$A$$0<\lambda_{0}\leq\lambda_{1}\leq\lambda_{2}\leq\cdots$$(e_{j})_{j}^{\infty_{=0}}$

(1-1)

Let

$E$

$Ae_{j}=\lambda_{j}e_{j}$$\delta\equiv(\sum_{j=0}^{\infty}\lambda_{j}^{-2})^{1/2}=\Vert A^{-1}\Vert_{HS}<\infty$.

be the standard CH-spaoe constructed from $(H, A)$ , that is, the$c\infty$-domain of

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$A$

equipped with the norms:(1-2)$| \xi|_{p}=|A^{p}\xi|_{0}=(\sum_{j=0}^{\infty}\lambda_{j}^{2p}\langle\xi,$$e_{j}\}^{2})^{1/2}$

,

Since
is of Hilbert-Schmidt type by assumption,and hence$A^{-1}$(1-3)$\xi\in E$$E$,$p\in$

R.

becomes a nuclear Fr\’echet space$E\subset H=L^{2}(T, \nu;R)\subset E^{*}$is also denoted bybecomes a Gelfand triple. The canonical bilinear form on
is always assumed to be equipped with the strong dual topology.
The dual space
By the Bochner-Minlos theorem there exists a unique probability measure on) such that(equipped with the Borel$E^{*}\cross E$$(\cdot, )$.

$E^{*}$$\mu$$E^{*}$$\sigma- field$(1-4)$\exp(-\frac{1}{2}|\xi|_{0}^{2})=\int_{E^{*}}e^{i\langle x,\xi)}\mu(dx)$,$\xi\in E$.

This is called the Gaussian measure and the probability spaceis called theGaussian space.
In a different context Gaussian space would mean merely a real (usually infinite dimensional) vector space equipped with Gaussian measure. In fact, -theory on Gaussian space is free not only from the particular construction of Gelfand triple (standard
CH-space) but also from the underlying space . However, those particular structures
together with the assumptions below are indispensable for our effective theory of distributions.
By construction each $\xi\in E$ is a function on determined up to v-null functions.
This hinders us from introducing a delta-function which is essential to our discussion.
Accordingly we are led to the following:(H1) For each $\xi\in E$ there exists a unique continuous function on such that $\xi(t)=$for v-a.e. $t\in T$ .
Once this is satisfied, we always assume that every element in is a continuous function
on and do not use the symbol . We further need:$\xi\in E$ , is continuous, i.e.,(H2) For each $t\in T$ a linear functional :;$t\in T$ , is continuous. (Recall that(H3) The map
carries the strong
dual topology.)
Under $(H1)-(H2)$ the convergence in implies the pointwise convergence as functions
on . If we have (H3) in addition, the convergence is uniform on every compact subset$(E^{*}, \mu)$$\mu$$L^{2}$

$T$

$T$

$T$

$\tilde{\xi}$$\sim\xi(t)$

$E$

$T$

$\sim\xi$$\delta_{t}$$E^{*}$$trightarrow\delta_{t}\in E^{*},$

$E$

$T$

$\delta_{t}\in E^{*}$$\xirightarrow\xi(t),$121
of . Moreover, it is noted that the properties $(H1)-(H3)$ are preserved under forming
tensor products. By another reason (see \S 4) we need one more assumption:$T$(S) $\lambda_{0}=\inf Spec(A)>1$ .

The constant number(1-5)$0<\rho\equiv\lambda_{0}^{-1}=\Vert A^{-1}\Vert_{oP}<1$

is important as well as defined in (1-1) to derive various inequalities, though we do
not use them explicitly in this paper.
$\delta$

2. Wiener-It\^o-Segal Isomorphism
For simplicity we put$(L^{2})=L^{2}(E^{*}, \mu;C)$.

In this section we recapitulate the famous Wiener-It\^o-Segal isomorphism between.
and the so-called Boson Fock space overis denoted byagain and its
The canonical bilinear form on
is also denoted by the same symbol. We now
bilinear extension to$\tau\in(E\otimesE)^{*}$
bydefine$(L^{2})$$H_{\mathbb{C}}$$(E^{\otimes n})^{*}\cross(E^{\otimes n})$$\langle\cdot, \rangle$$(E_{C}^{\otimes n})^{*}\cross(E_{C}^{\otimes n})$(2-1)$\langle\tau, \xi\otimes\eta\}=(\xi, \eta\rangle,$$\xi,$$\eta\in E$.

In other words,(2-2)$\langle\tau,$$\omega$) $= \int_{T}\omega(t, t)dt$ ,$\omega\in E\otimes E$.

follows from $(H1)-(H3)$ .
The fact that any
is a continuous function on
This distribution is called trace.
inductively as follows:
For $x\in E^{*}$ we define:$T\cross T$$\omega\in E\otimes E$$x^{\otimes n}:\in(E^{\otimes n})_{sym}^{*}$(2-3)$\{\begin{array}{l}x^{\otimes 0}\cdot.=lx^{\otimes l}\cdot.=xx^{\otimes n}.\cdot=x\otimes.\cdot x^{\otimes(n-l)}\cdot.-(n-l)\tau\otimes\cdot.x^{\otimes(n-2)}\wedge\wedge.\cdot\end{array}$$n\geq 2$

In other words, :(2-4)$x^{\otimes n}$: is defined as a unique element in$\{:x^{\otimes n}:,$$\xi^{\otimes n}\rangle=\frac{|\xi|_{0}^{n}}{2^{n/2}}H_{n}(\frac{\langle x,\xi)}{\sqrt{2}|\xi|_{0}})$,$(E^{\otimes n})_{sym}^{*}$$\xi\in E$,.

satisfying$\xi\neq 0$

,

122
where
denotes the Hermite polynomial of degree
by generating function:$H_{n}$(2-5)$\phi_{\xi}(x)\equiv\sum_{n=0}^{\infty}\{:x^{\otimes n}:,$$n$. Or equivalently, :,$\frac{\xi^{\otimes n}}{n!}\}=\exp(\langle x, \xi\rangle-\frac{1}{2}\langle\xi, \xi\rangle)$$x^{\otimes n}$$\xi\in E$: is defined.

Note that the right hand side of (2-5) is a “normalized” exponential function and the. We call an exponential vector.
identity is valid also for
The orthogonal relation of Hermite polynomials leads us to the following$\xi\in E_{C}$

(2-6)

,

$f_{m}\in E_{\mathbb{C}}^{\otimes m},$-approximation one can define a function$xrightarrow\langle:$ $x^{\otimes n}:,$$\int_{E^{*}}\{:$ $x^{\otimes m}:,$Then by usual$L^{2}$\langle :$\phi_{\xi}$$f_{m}\rangle$$g_{n}\rangle\mu(dx)=n!\langle f_{m}, g_{n}\rangle\delta_{mn}$$x^{\otimes n}:,$$g_{n}\wedge\in E_{\mathbb{C}}^{\otimes^{\wedge}n}$$f\rangle,$.

$x\in E^{*}$,

in -sense. Let
be the spaoe of all such functions. Then they
for any. Sinoe the polynomials, namely
become mutually orthogonal closed subspaces of, we come to the following
is dense in
the algebra generated by$f\in H_{C}^{\otimes n}\wedge$$L^{2}$$\mathcal{H}_{n}(C)$$(L^{2})$$(L^{2})$$\{\langle x, \xi\rangle ; \xi\in E_{C}\}$THEOREM 2.1 ( $WIENER- IT\hat{O}$ -SEGAL). The Hilbert spacesum decomposition:(2-7)$(L^{2})= \sum_{n=0}^{\infty}\oplus \mathcal{H}_{n}(C)$$(L^{2})$

admits an orthogonal.

More precisely, for each $\phi\in(L^{2})$ there exists a unique sequence$0,1,2,$, such that$f_{n}\in H_{C}^{\otimes n}\wedge,$$n=$$\cdots$(2-8)$\phi(x)=\sum_{n=0}^{\infty}\langle:x^{\otimes n}:,$where eachsum. In that case$\{:x^{\otimes n}:,$(2-9)$f_{n}\rangle$$f_{n}\rangle$,$x\in E^{*}$,

is a function in $?t_{n}(C)$ and the series is an orthogonal direct$|| \phi\Vert_{0}^{2}\equiv\int_{E^{*}}|\phi(x)|^{2}\mu(dx)=\sum_{n=0}^{\infty}n!|f_{n}|_{0}^{2}$

Furthermore, the above correspondence
tween
and the Boson Fock space over$\phirightarrow(f_{n})_{n=0}^{\infty}$$(L^{2})$$H_{C}$.

gives a unitary isomorphism be-.

According to the Wiener-It\^o decomposition (2-7) we define an operator $N$ by(2-10)$N\phi=n\phi$,$\phi\in \mathcal{H}_{n}(C)$,$n=0,1,2,$$\cdots$

123. ThisEquipped with the maximal domain, $N$ becomes a selfadjoint operator inoperator is called the number operator (because 7 $n(C)$ stands for the Hilbert spaoeof Bose particles in physical interpretation) and is one of the infinite dimensional
Laplacians on Gaussian space. We shall come back to this topic in \S 6.
$(L^{2})$$n$

3. Infinite Dimensional Rotation Group
Let $O(E;H)$ be the group of all linear homeomorphisms from $E$ onto itself preserving, namely 1 $g\xi|_{0}=|\xi|_{0}$ for $\xi\in E$ . In other words, $O(E;H)$ is the group of
the norm
automorphisms of the Gelfand triple $E\subset H\subset E^{*}$ . While, sinoe each $g\in O(E;H)$ isextended to an orthogonal operator on the Hilbert spaoe $H$ , we may regard $O(E;H)$ as
a subgroup of the full orthogonal group $O(H)$ . The group $O(E;H)$ is called the infinite).
dimensional rotation gmup (associated with the Gelfand triple
in an
The infinite dimensional rotation group $O(E;H)$ acts on the Gaussian spaoe
obvious manner:$|\cdot|_{0}$$E\subset H\subset E^{*}$$E^{*}$(3-1)$\langle x, g\xi\rangle=\langle g^{*}x, \xi\rangle$,$x\in E^{*}$,$\xi\in E$.

As is seen immediately from (1-4), the characteristic functional of is invariant under
the action of $O(E;H)$ . Henoe the uniqueness of a characteristic functional implies that$x\in E^{*},$ $g\in O(E;H)$ .
the Gaussian measure is invariant under the action:
We then come to a natural unitary representation of $O(E;H)$ on$\mu$$xrightarrow g^{*}x,$$\mu$$(L^{2})$(3-2)$(\Gamma(g)\phi)(x)=\phi(g^{*}x)$

As is easily verified, if(3-3)

Henoe each$\phi\in(L^{2})$,$\phi\in(L^{2})$$g\in O(E;H)$.

is expressed as in (2-8), we have$( \Gamma(g)\phi)(x)=\sum_{n=0}^{\infty}\langle:x^{\otimes n}:,$$\mathcal{H}_{n}(C)$,$g^{\otimes n}f_{n}\}$.

in the Wiener-It\^o decomposition is an invariant subspace. Moreover,is an irreducible
THEOREM 3.1. The Wiener-It\^o decomposition
decomposition of the unitary representation
of $O(E;H)$ . Furthermore, all$n(C)$are mutually inequivalent.
irreducible subspaces 7$(L^{2})=\Sigma_{n=0}^{\infty}\oplus \mathcal{H}_{n}(C)$$(\Gamma, (L^{2}))$This is a simple consequenoe of the following fundamental result.
THEOREM 3.2 (UMEMURA [27]). Let be a symmetric operator on $(L^{2})=L^{2}(E^{*}, \mu)$which is invariant under the action of $O(E;H)_{f}$ and assume that for any $\xi\in E,$
belongs to the domain of . Then can be expressed as a jfunction of $N$ .
$\Xi$$e^{i\langle\cdot,\xi\rangle}$

$\Xi$

$\Xi$

For the precise meaning of “a function of $N$ ’ see the original paper. Instead we notebe a selfadjoint operator onwith domain
here a significant consequence: Let$\Xi$$(L^{2})$

124
containing all exponential functions of the form
Dom
$O(E;H)$, then there exists a real sequenoe
under$(\Xi)$$e^{i\langle\cdot,\xi\rangle},$$(\alpha_{n})_{n=0}^{\infty}$$\phi\in Dom(\Xi)\cap \mathcal{H}_{n}(C)$.

$\xi\in E$. If is invariant

such that$\Xi$$\Xi\phi=\alpha_{n}\phi$for

The irreducible representations mentioned in Theorem 3.1 are characterized by Matsushima, Okamoto and Sakurai [16] and by Okamoto and Sakurai [25].

4. White Noise Functionals, where is the same operator as we
We first need a second quantized operator
used in \S 1 to construct the Gelfand triple $E\subset H=L^{2}(T, \nu;R)\subset E^{*}$ and the Gaussianspaoe. Suppose thatis given as$A$$\Gamma(A)$$\phi\in(L^{2})$$(E^{*}, \mu)$(4-1)$\phi(x)=\sum_{n=0}^{\infty}\langle:x^{\otimes n}:,$$f_{n}\rangle$

according to the Wiener-It\^o-Segal isomorphism. We then put(4-2)$\Gamma(A)\phi(x)=\sum_{n=0}^{\infty}\langle:x^{\otimes n}:,$$A^{\otimes n}f_{n}\}$.

In the previous section we employed the same symbol for a particular unitary representation of $O(E;H)$ . However, there will occur no confusion due to the fact (3-3). It
equipped with the maximal domain becomes a positive selfadjoint
is known that
operator on.
Let $(E)$ be the standard CH-spaoe constructed from the pair $((L^{2}), \Gamma(A))$ . Since
admits Hilbert-Schmidt inverse by the hypothesis (S) in \S 1, $(E)$ is a nuclear
Ilir\’echet spaoe and$\Gamma$$\Gamma(A)$$(L^{2})$$\Gamma(A)$(4-3)$(E)\subset(L^{2})=L^{2}(E^{*}, \mu;C)\subset(E)^{*}$

becomes a complex Gelfand triple. Elements in $(E)$ andare called a test (white
noise) functional and a genemlized (white noise) functional, respectively. We denote by\langle \langle .,the canonical bilinear form on $(E)^{*}\cross(E)$ and by
the norm induced from, namely,$(E)^{*}$$\cdot\rangle\rangle$$\Vert\cdot\Vert_{p}$$\Gamma(A)$(4-4)$|| \phi||_{p}^{2}=\Vert\Gamma(A)^{p}\phi\Vert_{0}^{2}=\sum_{n=0}^{\infty}n!|(A^{\otimes n})^{p}f_{n}|_{0}^{2}=\sum_{n=0}^{\infty}n!$I$f_{n}|_{p}^{2}$,$\phi\in(E)$,where andare related as in (4-1). This identity is compatible with (2-9).
By construction each $\phi\in(E)$ is defined only up to -null functions. However, it
follows from Kubo-Yokoi’s continuous version theorem [12] that for $\phi\in(E)$ the right$\phi$$(f_{n})_{n=0}^{\infty}$

$\mu$

125
hand side of (4-1) converges absolutely at each $x\in E^{*}$ and becomes a unique continuous. $x\in E^{*}$ . Thus, $(E)$ is always assumed
which coincides withforfunction on
and for $\phi\in(E)$ the right hand side of (4-1)
to be a spaoe of continuous functions on.
is understood as pointwisely convergent series as well as in the sense of norms
For a generalized white noise functional $\Phi\in(E)^{*}$ there exists a unique sequence, such that$n=0,1,2,$$E^{*}$$\phi(x)$$\mu- a.e$$E^{*}$$||\cdot||_{p}$$F_{n}\in(E_{C}^{\otimes n})_{sym}^{*},$$\cdots$(4-5),$\langle\langle\Phi, \phi\rangle\rangle=\sum_{n=0}^{\infty}n!\langle F_{n}, f_{n}\rangle$

for $\phi\in(E)$ given as in (4-1). In that case it holds that
(4-6)$|| \Phi\Vert_{-p}^{2}=\sum_{n=0}^{\infty}n!|F_{n}|_{-p}^{2}$.

This is finite for all sufficiently large $p\geq 0$ and is compatible with (4-4). It is thenconvenient to adopt a formal expression:(4-7)$\Phi(x)=\sum_{n=0}^{\infty}\langle:x^{\otimes n}:,$$F_{n}\rangle$.

Conversely, we agree that (4-7) defines a generalized white noise functional via (4-5)
for some $p\geq 0$ .
wheneverThe simplest example of generalized white noise functionals would be white noise. For simplicity
belongs tocoordinate. For each $t\in T,$ $\Phi_{t}(x)=$we put$\Phi$$\sum_{n=0}^{\infty}n!|F_{n}|_{-p}^{2}<\infty$$\langle:(4-8)$(E)^{*}$

x:, \delta_{t}\rangle=\langle x, \delta_{t}\rangle$$x(t)=\langle x,$$\delta_{t}$),$t\in T$,

which may be regarded as white noise analogue of the usual coordinate $(x_{1}, \cdots x_{D})$ of.
Euclidean spaoe
As for exponential vectors (2-5) we remind the following$R^{D}$PROPOSITION 4.1.
subspace of $(E)$ .

$\phi_{\xi}\in(E)$for any$\xi\in E_{C}$and such exponential vectors span a dense

The S-tmnsform of $\Phi\in(E)^{*}$ is a function on(4-9)$S\Phi(\xi)=\langle\langle\Phi,$$E_{\mathbb{C}}$defined by,$\phi_{\xi}\rangle\}=e^{-\langle\xi,\xi\rangle/2}\int_{E^{*}}\Phi(x)e^{\langle x,\xi\rangle}\mu(dx)$$\xi\in E_{C}$.

126
While, the T-tmnsform is defined by(4-10)$T\Phi(\xi)=\{\langle\Phi,$$e^{i\langle\cdot,\xi\rangle} \rangle\rangle=\int_{E^{*}}\Phi(x)e^{i(x,\xi)}\mu(dx)$,$\xi\in E_{C}$.

Of course the integral expressions are valid only when the integrands are integrable
functions, in paticular when $\Phi\in(E)$ . There is a simple relation:(4-11)$T\Phi(\xi)=S\Phi(i\xi)e^{-\langle\xi,\xi)/2}$,$S\Phi(\xi)=T\Phi(-i\xi)e^{-(\xi,\xi\rangle/2}$,$\xi\in E_{\mathbb{C}}$.

5. Integral Kernel Operators and Fock Expansion
For $y\in E^{*}$ we put(5-1)$D_{y} \phi(x)=\lim_{\thetaarrow 0}\frac{\phi(x+\theta y)-\phi(x)}{\theta}$,$x\in E^{*}$,$\phi\in(E)$.

becomes a continuous operator (in fact
It is known that the limit always exists and, wea derivation) from $(E)$ into itself, i.e., $D_{y}\in \mathcal{L}((E), (E))$ . In particular, forput$D_{y}$$y=\delta_{t}$

(5-2)

,

$\partial_{\ell}=D_{\delta_{t}}$$t\in T$.

In most physical literature
is called an annihilation opemtor at point $t\in T$ and isunderstood to be (unbounded) operator-valued distribution. However, in our setup
is therefore
is just a continuos opemtor on $(E)$ for itself. The adjoint
called creation opemtor. Note also that is often called Hida’s differential operator as
well. That we are free from smeared creation and annihilation operators is one of themost significant features of white noise calculus.
The annihilation and creation operators satisfy the canonical commutation relation
in a generalized sense:$\partial_{t}$$\partial_{t}$$\partial_{t^{*}}\in \mathcal{L}((E)^{*}, (E)^{*})$$\partial_{t}$(5-3)$[\partial_{s}, \partial_{t}]=0$,$[\partial_{s}^{*}, \partial_{t^{*}}]=0$,$[\partial_{s}, \partial_{t^{*}}]=\delta_{s}(t)$.

The precise meaning of the last identity is:(5-4)$[D_{y}, D_{\xi}^{*}]=\langle y, \xi\rangle I$

Note that bothandbelong tosee (5-7) and Theorem 5.1 below.
$D_{y}$$D_{\xi}^{*}$,$y\in E^{*}$$\mathcal{L}((E), (E))$,$\xi\in E$.

and their compositions are meaningful,

127
With each
expression is given by$\kappa\in(E_{\mathbb{C}}^{\otimes(l+m)})^{*}$(5-5)

we may associate an integml kernel opemtor whose formal$–l,m \int_{T^{l+m}}\kappa(s_{1}, \cdots s_{l},t_{1}, \cdots t_{m})\partial_{s_{1}}^{*}\cdots\partial_{s_{l}}^{*}\partial_{t_{1}}\cdots\partial_{t_{m}}ds_{1}\cdots ds_{l}dt_{1}\cdots dt_{m}$,where is called the kernel distribution. More presicely, it is defined through two
canonical bilinear forms:$\kappa$(5-6)$\langle\{--(\kappa)\phi,$$\psi\rangle\rangle=\langle\kappa,$$\langle\langle\partial_{s_{1}}^{*}\cdots\partial_{s_{l}}^{*}\partial_{t_{1}}\cdots\partial_{t_{m}}\phi,$,$\psi\rangle\rangle\rangle$$\phi,\psi\in(E)$.

It is proved $that–(\kappa)\in \mathcal{L}((E), (E)^{*})$ , see [8] for further details. For example,(5-7)$–0,1 \int_{T}y(t)\partial_{t}dt=D_{y}$TnEOREM 5.1 ([8]). Let$\kappa\in(E_{\mathbb{C}}^{\otimes(l+m)})^{*}$$\kappa\in(E_{C}^{\otimes l})\otimes(E_{C}^{\otimes m})^{*}\cong \mathcal{L}(E_{\mathbb{C}}^{\otimes m}, E_{C}^{\otimes l})$.

,$y\in E^{*}$.

$Then—\iota_{m}(\kappa)\in \mathcal{L}((E), (E))$.

if and only if

is symmetric
By virtue of (5-3) we may assume that the kernel distribution$m$with respect to the first and the lastvariables independently. We denote by
the spaoe of such distributions. The importanoe of an integral kernel
operator is due to the following$\kappa$$l$$(E_{\mathbb{C}}^{\otimes(l+m)})_{sym(l,m)}^{*}$

THEOREM 5.2 ([20], [21], [22]). For anyof kernel distributions$\Xi\in \mathcal{L}((E), (E)^{*})$$\kappa_{l,m}\in(E_{C}^{\otimes(l+m)})_{sym(l,m)}^{*}$(5-8)$\Xi\phi=\sum_{l,m=0}^{\infty}--l,m$there exists a unique familysuch that$\phi\in(E)$’,. Moreover, if $\Xi\in \mathcal{L}((E), (E))$ , thenwhere the right hand side converges inand the infinite series (5-8) converges in $(E)$ .
$(E)^{*}$$\kappa_{l,m}\in$$((E_{C}^{\otimes l})\otimes(E_{C}^{\otimes m})^{*})_{sym(l,m)}$

The unique expression of
expansion of and denoted simply by$\Xi\in \mathcal{L}((E), (E)^{*})$

given in Theorem 5.2 is called the Fock

$\Xi$

(5-9)

$\Xi=\sum_{l,m=0}^{\infty}---\iota_{m}(\kappa_{l,m})$.

A few simple examples will be found in the rest of the paper, for further discussion seee.g., [20], [21], [22].

128
For

$\Xi\in \mathcal{L}((E), (E)^{*})$

a function on(5-10)$-\wedge--(\xi, \eta)=\langle\{\Xi\phi_{\xi},$

is called the symbol of(5-11)$\Xi$

defined by$E_{C}\cross E_{C}$,$\phi_{\eta}\rangle\}$. For example, for$e^{-\langle\xi,\eta\rangle_{-}^{\underline{\underline{\wedge}}}}( \xi,\eta)=\sum_{1,m=0}^{\infty}\langle\kappa\iota_{m},$Hence, in order to find kernel distributions
only to compute the Taylor expansion of$\Xi$$\xi,$$\eta\in E_{C}$,with Fock expansion (5-9) we have,$\eta^{\otimes l}\otimes\xi^{\otimes m}\rangle$$\kappa_{l,m}$$\xi,$$\eta\in E_{\mathbb{C}}$.

from a given $\Xi\in((E), (E)^{*})$ one need

$e-\langle\xi,\eta\rangle_{-}^{\wedge}--(\xi, \eta)$.

6. Infinite Dimensional Laplacians
Consider the following two integral kernel operators:(6-1)$\Delta_{G-0,2}^{-}=-(\tau)=\int_{TxT}\tau(s, t)\partial_{s}\partial_{t}dsdt=\int_{T}\partial_{t^{2}}dt$

,

,

(6-2)

$N=—1,1( \tau)=\int_{T\cross T}\tau(s, t)\partial_{s}^{*}\partial_{t}dsdt=\int_{T}\partial_{t^{*}}\partial_{t}dt$

where
was defined in (2-1). (It is easily verified that $—1,1(\tau)$ coincides
with $N$ introduced in \S 2.) The operators
and $N$ are called the Gmss Laplacianand the number operator, respectively. By Theorem 5.1 bothand $N$ belong to. In fact,sinoe the corresponding operator under the isomorphismis the identity. Note that$\tau\in(E\otimes E)_{sym}^{*}$$\Delta_{G}$$\Delta_{G}$$\mathcal{L}((E), (E))$$\tau\in E\otimes E^{*}$$\mathcal{L}(E, E)\cong E\otimes E^{*}$(6-3)$\Delta_{G-2,0}^{*-}=-(\tau)$

and that the number operator is symmetric, i.e.,$(E)^{*}$

is a continuous extension of $N$ to.

The action of the above operators on
easily derived:$\psi_{n}(x)=\langle:x^{\otimes n}:,$$N\psi_{n}=N^{*}\psi_{n}=n\psi_{n}$(6-4)$N^{*}$$\Delta_{G}\psi_{n}=n(n-1)(\xi,$$\xi^{\otimes n}\rangle,$$\xi\in E_{C},$$n=0,1,$$\cdots$is,$\xi\rangle$$\psi_{n-2}$$(\Delta_{G}^{*}\psi_{n})(x)=\{:x^{\otimes(n+2)}:,$,$\tau\otimes\xi^{\otimes n}\rangle$.

It is also possible to express these Laplacians in terms of discrete coordinate. For
simplicity we put(6-5)$D_{j}=D_{e_{j}}$,$j=0,1,2,$$\cdots$.

129
PROPOSITION 6.1. For $\phi\in(E)$ we have
(6-6)$\Delta_{G}\phi=\sum_{j=0}^{\infty}D_{j}^{2}\phi$,$N \phi=\sum_{j=0}^{\infty}D_{j}^{*}D_{j}\phi$,where the right hand sides converge in $(E)$ .
By the above result together with the definitions we might be convinced that bothand $N$ are white noise analogies of a finite dimensional Laplacian.
. It is known [11] that.theWe then find a relation between two Laplacians $N$ andpointwise multiplication gives rise to a continuous bilinear map from $(E)\cross(E)arrow(E)$ .
$\phi\in(E)$ ,Henoe each $\Phi\in(E)^{*}$ is identified with multiplication operator
by$\Delta_{G}$$\Delta_{G}$$\phiarrow\Phi\phi=\phi\Phi,$

(6-7)

Thus

$\langle\langle\Phi\phi, \psi\rangle\rangle=\langle\langle\Phi, \phi\psi\rangle\rangle$$\Phi\in \mathcal{L}((E), (E)^{*})$. Moreover,,$\phi,$$\Phi\in \mathcal{L}((E), (E))$$\psi\in(E)$.

if and only if $\Phi\in(E)$ .

LEMMA 6.2. It holds that
(6-8)

$D_{j}+D_{j}^{*}=\langle x, e_{j}\rangle$,

where the right hand side is identified with multiplication opemtor.
PROOF. For$\xi,$$\eta\in E_{\mathbb{C}}$we have$\langle\langle(D_{j}+D_{j}^{*})\phi_{\xi},$$\phi_{\eta}\rangle\rangle=\langle\langle D_{j}\phi_{\xi},$$\phi_{\eta}\rangle\}+\langle\langle\phi_{\xi},$$D_{j}\phi_{\eta}\rangle\rangle$$=(\langle e_{j}, \xi\rangle+\langle e;, \eta))\langle\langle\phi_{\xi},$$=\langle e_{j}, \xi+\eta\rangle e^{\langle\xi,\eta\rangle}$

On the other hand, sinoe$\langle\{\{x,$$e_{j}$)$\phi_{\xi},$$\phi_{\xi}\phi_{\eta}=\phi_{\xi+\eta}e^{\langle\xi,\eta\rangle}$$\phi_{\eta}\rangle\rangle=\langle\{\langle x, e_{j}\rangle,$$\phi_{\eta}\rangle\rangle$.

we see that$\phi_{\xi}\phi_{\eta}\rangle\rangle=\{\langle\langle x, e_{j}\rangle,$$\phi_{\xi+\eta}\rangle\}e^{(\xi,\eta\rangle}=\langle e_{j},$$\xi+\eta$)$e^{(\xi,\eta\rangle}$.

Combining the above two expressions, we come to$\langle\langle(D_{j}+D_{j}^{*})\phi_{\xi},$$\phi_{\eta}\rangle\rangle=\langle\langle\{x,$Then the assertion follows from Proposition 4.1.

$e_{j}\rangle$$\phi_{\xi},$$\phi_{\eta}\rangle\rangle$

.
QED

130
PROPOSITION 6.3. It holds that
(6-9)

where

$-N= \Delta_{G}-\sum_{j=0}^{\infty}\langle x,$$\langle x, e_{j}\rangle$$e_{j}$)$D_{j}$,

is regarded as multiplication op emtor.

This is an immediate consequence of Proposition 6.1 and Lemma 6.2.

7. Rotation-invariant Operators
The main purpose of this section is to characterize all rotation-invariant operators bymeans of Fock expansion (Theorem 5.2), though the result was first proved in [19] using
a weaker form of Fock expansion.
is rotation-invariant if
We say that$\Xi\in \mathcal{L}((E), (E)^{*})$(7-1)$\Gamma(g)^{*}\Xi\Gamma(g)=\Xi$

for all $g\in O(E;H)$ .

Note here that $\Gamma(g)\in \mathcal{L}((E), (E))$ . By definition, if is rotation-invariant, so is
The condition (7-1) for $\Xi\in \mathcal{L}((E), (E))$ is equivalent to the following
$\Xi$

(7-2)

$\Gamma(g)\Xi=\Xi\Gamma(g)$$\Xi^{*}$.

for all $g\in O(E;H)$ ,which is the usual rotation-invariance.
It is rather straightforward to see that $N$ andare rotation-invariant. (This fact
follows from Lemma 7.4 and Proposition 7.5 below.) Therefore
is also rotation$N$
invariant, whileis the extension of . The goal of this section is the following
significant characterization of rotation-invariant operators.
$\Delta_{G}$$\Delta_{G}^{*}$$N^{*}$

THEOREM 7.1. Let
and let $\Xi=\sum_{l},-(\kappa_{1,m})$ be its Fock expansion. Then is rotation-invariant if and only if $all–(\kappa_{l,m})$ are rotation-invariant.
$\Xi\in \mathcal{L}((E), (E)^{*})$$\Xi$

THEOREM 7.2. Let
and assume $that–(\kappa)$ is rotation-invariant. If$l+m$ is odd, $then^{-}-(\kappa)=0$ . If $l+m$ is even,is a linear combination of
being non-negative integers such that $\alpha+\beta+\gamma\leq(l+m)/2$ .
with$\kappa\in(E_{\mathbb{C}}^{\otimes(l+m)})^{*}$$then—\iota_{m}(\kappa)$$(\Delta_{G}^{*})^{\alpha}N^{\beta}\Delta_{G}^{\gamma}$$\alpha,$$\beta,$$\gamma$

THEOREM 7.3. Let
and assume $that^{-}--l,m(\kappa)$ is rotation-invariant.
$then^{-}--\iota_{m}(\kappa)=0$. If $l+m$ is even,is a linear combination
If $l+m$ is odd,
being non-negative integers such that $\beta+\gamma\leq(l+m)/2$ .
withof$\kappa\in(E_{c}^{\otimes l})\otimes(E_{\mathbb{C}}^{\otimes m})^{*}$$then—\iota_{m}(\kappa)$$N^{\beta}\Delta_{G}^{\gamma}$$\beta,$$\gamma$In other words, any rotation-invariant operator
is generated by
and $N$ , and any rotation-invariant operator $\Xi\in \mathcal{L}((E), (E))$ is generated by
and $N$ . It is also easily checked that$\Xi\in \mathcal{L}((E), (E)^{*})$$\Delta_{G}$(7-3)$[\Delta_{G}, N]=2\Delta_{G}$.

$\Delta_{G}^{*}$,$\Delta_{G}$131
Henoe any product ofand $N$ (whenever it is well defined on $(E)$ ) may be
being non-negative integers. This iswithrearranged as a sum ofalso related to the normal ordering of creation and annihilation operators.
PROOF OF THEOREM 7.1. Suppose we are given
with Fock expansion$\Delta_{G}^{*},$$\Delta_{G}$$(\Delta_{G}^{*})^{\alpha}N^{\beta}\Delta_{G}^{\gamma}$$\alpha,$$\beta,\gamma$$\Xi\in \mathcal{L}((E), (E)^{*})$

(7-4)

where
(7-5)

$\Xi=\sum_{1,m=0}^{\infty}---\iota_{m}(\kappa_{1,m})$$\kappa_{l,m}\in(E_{\mathbb{C}}^{\otimes(l+m)})_{sym(l,m)}^{*}$. Sinoe,for$\Gamma(g)\phi_{\xi}=\phi_{g\xi}$$(\Gamma(g)^{*}\Xi\Gamma(g))^{\sim}(\xi, \eta)=\langle\langle\Xi\Gamma(g)\phi_{\xi},$$=\langle\langle\Xi\phi_{g\xi},$$\xi\in E_{C}$, we obtain$\Gamma(g)\phi_{\eta}\rangle\rangle$$\phi_{g\eta}\rangle\rangle$$=_{-}(g\xi,g\eta)\underline{\underline{\wedge}}$,$\xi,$$\eta\in E_{C}$.

Moreover, from (5-10) we see that
(7-6)$- \underline{\underline{\wedge}}(g\xi,g\eta)=e^{\langle g\xi,g\eta\rangle}\sum_{l,m=0}^{\infty}\langle\kappa_{l,m},$$(g\eta)^{\otimes l}\otimes(g\xi)^{\otimes m}\rangle$$=e^{\langle\xi,\eta\}} \sum_{l,m=0}^{\infty}\langle(g^{\otimes(l+m)})^{*}\kappa,$$\eta^{\otimes l}\otimes\xi^{\otimes m}\rangle$.

It then follows from (7-5) and (7-6) that$\Gamma(g)^{*}\Xi\Gamma(g)=\sum_{l,m=0}^{\infty}--l,m-((g^{\otimes(l+m)})^{*}\kappa_{l,m})$is the Fock expansion. In particular,$\Gamma(g)^{*-}--l,m(\kappa)\Gamma(g)=--l,m-((g^{\otimes(1+m)})^{*}\kappa)$.

It then follows from the uniqueness of the Fock expansion that is rotation-invariant
if and only $if–(\kappa_{l,m})$ is rotation-invariant for all $m=0,1,2,$ .
QED
We say thatis mtation-invariant if $(g^{\otimes n})^{*}F=F$ for all $g\in O(E;H)$ .
During the proof of Theorem 7.1 we have established the following
$\Xi$

$l,$

$\cdots$$F\in(E_{C}^{\otimes n})^{*}$LEMMA 7.4. Let
if is rotation-invariant.

$\kappa\in(E_{c}^{\otimes(l+m)})_{sym(l,m)}^{*}$. $Then–(\kappa)$ is rotation-invariant if and only$\kappa$

Thus the proofs of Theorems 7.2 and 7.3 are essentially reduced to listing up therotation-invariant distributions. The full list is, in fact, described satisfactorily as below,
though the long combinatorial proof is omitted, see [19].

132
is mtation-invariant. If is odd, then
PROPOSITION 7.5. Assume that.
$F=0$ . If is even, say $n=2m$ , then $F$ is a linear combination of
Moreover, the dimension of rotation-invariant distributions inis $(n-1)!!$ .
$F\in(E_{C}^{\otimes n})^{*}$$n$$(\tau^{\otimes m})^{\sigma},$$n$$\sigma\in \mathfrak{S}_{n}$$(E_{C}^{\otimes n})^{*}$

Here is notation. For$F\in(E_{\mathbb{C}}^{\otimes n})^{*}$$\langle F^{\sigma}, \xi_{1}\otimes\cdots\otimes\xi_{n}\rangle=\langle F,$

and$\sigma\in \mathfrak{S}_{n}$we define$\xi_{\sigma(1)}\otimes\cdots\otimes\xi_{\sigma(n)}\rangle$,$F^{\sigma}$by$\xi_{1},$$\cdots\xi_{n}\in E_{\mathbb{C}}$.

PROOF OF THEOREM 7.2. Let
isand suppose thatrotation-invariant. Then, is rotation-invariant by Lemma 7.4. If $l+m$ is odd, it.
follows from Proposition 7.5 that $\kappa=0$ and henoe$l+m$
We next consider the case whenis even. It follows again from Proposition 7.5. For eachthat is a linear combination of
we mayfindsuch that$\kappa\in(E_{C}^{\otimes(l+m)})_{sym(l,m)}^{*}$$—\iota_{m}(\kappa)$$\kappa$$—\iota_{m}(\kappa)=0$$(\tau^{\otimes(l+m)/2})^{\sigma},$$\kappa$$\sigma\in \mathfrak{S}_{l+m}$$\sigma\in \mathfrak{S}_{l+m}$$\sigma’\in \mathfrak{S}_{l}\cross \mathfrak{S}_{m}$$(\tau^{\otimes(1+m)/2})^{\sigma\sigma’}$$= \sum e_{1_{1}}^{\otimes 2}\otimes\cdots\otimes e_{1_{\alpha}}^{\otimes 2}\otimes e_{j_{1}}\otimes\cdots\otimes e_{j_{\beta}}\otimes e_{j_{1}}\otimes\cdots\otimes e_{j_{\beta}}\otimes e_{k}^{\otimes_{1}2}\otimes\cdots\otimes e_{k_{\gamma}}^{\otimes 2}$$=\tau^{\otimes\alpha}\otimes\lambda_{\beta}\otimes\tau^{\otimes\gamma}$

for some non-negative integers$\alpha,$$\beta,\gamma$

with $2\alpha+\beta=l$ and $2\gamma+\beta=m$ , where$\lambda_{\beta}=\sum_{j_{I},\cdots,j_{\beta}=0}^{\infty}e_{j_{1}}\otimes\cdots\otimes e_{j_{\beta}}\otimes e_{j_{1}}\otimes\cdots\otimes e_{j_{\beta}}$.

In view of (6-1) we have$\Delta_{G-0,2(\tau)=\sum_{j=0}^{\infty}-0,2}^{-}=--$,$\Delta_{G-2}^{*-}=-,0(\tau)=\sum_{j=0}^{\infty}---2,0(e_{j}\otimes e_{j})$.

Then a straightforward computation implies that$-l,m-\otimes(l+m)/2\sigma--l,m\otimes(1+m)/2--\rho,\rho(\lambda_{\beta})\Delta_{G}^{\gamma}$

Note$that—\rho,\rho(\lambda_{\beta})$

is a polynomial of the number operator $N$ of degree , in fact,$\beta$$—\rho,\rho(\lambda_{\beta})=N(N-1)\cdots(N-(\beta-1))$

is a linear combination of$(l+m)/2$ and therefore, so is
.

Henoe

.

$—\iota_{m}((\tau^{\otimes(l+m)/2})^{\sigma})$$—\iota_{m}(\kappa)$.

$(\Delta_{G}^{*})^{\alpha}N^{\beta}\Delta_{G}^{\gamma}$

with$\alpha+\beta+\gamma\leq$

QED

133
For the proof of Theorem 7.3, taking Theorem 7.2 into account, we only need to show
does not appear if $\Xi\in \mathcal{L}((E), (E))$ . But this is an easy consequenoe of thethat.
for any $\phi\in(E)$ withfact (see (6-4)) that$g\in O(E;H)$ ,By Theorem 3.2 any bounded operator oncommuting with all$N$is a function of the number operator . However, it is clear that the Gross Laplacianis not a function of $N$ , see e.g., (6-4). In other words, the Gross Laplacian can not be. This is also illustrated bygrasped whenever we restrict ourselves to operators on
the fact thaton its proper -domain, i.e., on the spaoe of all $\phi\in(E)$ with$\Delta_{G}^{*}$$\phi\neq 0$$\Delta_{G}^{*}\phi\not\in(E)$$(L^{2})$$\Gamma(g),$$(L^{2})$$L^{2}$$\Delta_{G}^{*}=0$$\Delta_{G}^{*}\phi\in(L^{2})$.

We here recall Umemura’s heuristic argument of deriving the number operator from fishould
nite dimensional Laplacians. First the finite dimensional Laplacian
be modified using the mapping
which is a unitary isomorphism$L^{2}(R^{D},dx)$-spaoefromonto thewith Gaussian measure. The resultant
over
expression is;$\sum_{j=1}^{D}\partial^{2}/\partial x_{j}^{2}$$\phirightarrow(2\pi)^{D/2}e^{|x|^{2}/4}\phi$$L^{2}$(7-7)$R^{D}$$\sum_{j=1}^{D}(\frac{\partial^{2}}{\partial x_{j}^{2}}-x_{j}\frac{\partial}{\partial x_{j}}+\frac{x_{j}^{2}}{4}-\frac{1}{2})$.

(is a projective limit ofwith Gaussian measure.) Then, taking the “convergent
terms,” Umemura defined an infinite dimensional Laplacian by$R^{D}$$E^{*}$(7-8)$\Delta=\sum_{j=1}^{\infty}(\frac{\partial^{2}}{\partial x_{j}^{2}}-x_{j}\frac{\partial}{\partial x_{j}})$.

This operator acts on cylindrical functions of the form:$\phi(x)=f(\langle x, e_{1}\rangle, \cdots \langle x, e_{n}\rangle)$,$x\in E^{*}$,. It is then easily verified that $\Delta=-N$ . In fact, (7-8) is comparable
where
to (6-9).
Umemura [27] showed that $N$ (or equivalently ) is the essentially unique rotationinvariant operator. However, as was shown above, within white noise calculus $N$ isdecomposed into two rotation-invariant operators. Furthermore, we shall see that the“divergent terms” in (7-7) involves another rotation-invariant operator. Consider white
noise analogue of the Euclidean norm:$x_{j}=\langle x, e_{j}\rangle$$\Delta$$R(x)=\langle:x^{\otimes 2}:,$$\tau\rangle$.

134This is a generalized white noise functional (see \S 4) and admits another expression:$R(x)= \sum_{j=0}^{\infty}\langle:x^{\otimes 2}:,$$e_{j}\otimes e_{j\rangle}$$= \sum_{j=0}^{\infty}(\{x\otimes x, e_{j}\otimes e_{j}\rangle-\langle\tau, e_{j}\otimes e_{j}))$$= \sum_{j=0}^{\infty}(\langle x, e_{j}\rangle^{2}-1)$.

Then, it is apparent that $R$ is involved in the “divergent terms” of (7-7). Moreover, as
multiplication operator, $R$ is related to the Laplacians:$R=2N+\Delta_{G}+\Delta_{G}^{*}$.

We have thus observed an interesting contrast between rotation-invariant operators on
white noise functionals and those on a finite dimensional Euclidean space.

8. Regular One-parameter Subgroups
We begin with general notion. Let be a nuclear Fr\’echet space with defining Hilbert, taking $X=E$ or
ian seminorms
into consideration. Let $GL(X)$ be
the group of all linear homeomorphisms from SC onto itself. A one-parameter subgroupis called differentiable if$\mathfrak{X}$$\mathfrak{X}=(E)$$\{||\cdot\Vert_{\alpha}\}_{\alpha\in A}$$\{g_{\theta}\}_{\theta\in R}\subset GL(\mathfrak{X})$(8-1)$X \xi=\lim_{\thetaarrow 0}\frac{g_{\theta}\xi-\xi}{\theta}$

converges in ec for any
X. In that casebecomes a linear operator from ec into
and,usual,
itselfas.
is called the infinitesimal generator of
It is known that a subset of a nuclear space is compact if and only if it is closed andbounded. Then simple application of the Banach-Steinhaus theorem leads us to the$X$$\xi\in$$\{g_{\theta}\}_{\theta\in R}\subset GL(\mathfrak{X})$following
LEMMA 8.1. Let
be a differentiable one-parameter subgroup. Then$X$
its infinitesimal genemtor
is always continuo us, $i.e$ ., $X\in \mathcal{L}(X, X)$ . Moreover, theconvergence (8-1) is uniform on every compact (or equivalently, bounded) subset of X,
namely,$\{g_{\theta}\}_{\theta\in R}\subset GL(\mathfrak{X})$(8-2)$\lim_{\thetaarrow 0}\sup_{\xi\in K}\Vert\frac{g_{\theta}\xi-\xi}{\theta}-X\xi\Vert_{\alpha}=0$

135
for any$\alpha\in \mathcal{A}$and any compact (or bounded) subset $K\subset X$ .

By a standard argument one may prove the uniqueness of an infinitesimal generator
of a differentiable one-parameter subgroup. However, in general, not every
can be an infinitesimal generator of a differentiable one-parameter subgroup of $GL(X)$ .
We give here a sufficient condition.
$X\in \mathcal{L}(X, X)$

PROPOSITION 8.2. Let
and assume that there exists $r>0$ such thatis equicontinuous, namely, for every
there exist $C=C(\alpha)\geq 0$
andsuch that$X\in \mathcal{L}(\mathfrak{X}, X)$$\{(rX)^{n}/n!\}_{n=0}^{\infty}$$\alpha\in \mathcal{A}$$\beta=\beta(\alpha)\in \mathcal{A}$$\sup_{n\geq 0}\frac{1}{n!}\Vert(rX)^{n}\xi||_{\alpha}\leq C||\xi\Vert_{\beta}$,$\xi\in X$

Then there exists a differentiable one-pammeter subgroupfinitesimal generator $X$ .

.

$\{g_{\theta}\}_{\theta\in R}\subset GL(X)$

In that case we observe a stronger property than stated in (8-2): for any
existssuch thatwith in-$\alpha\in A$

there$\beta\in \mathcal{A}$$\lim_{\thetaarrow 0_{||}}\sup_{\xi||_{\beta}\leq 1}\Vert\frac{g_{\theta}\xi-\xi}{\theta}-X\xi\Vert_{\alpha}=0$.

is called regular. AlSuch a differentiable one-parameter subgroup
though it is not yet clear whether the notion of a regular one-parameter subgroup plays
an essential role in white noise calculus, we feel it practically useful.
Here are simple examples in case of $X=(E)$ .
EXAMPLE 8.3 (TRANSLATION OPERATOR). For $y\in E^{*}$ We put$\{g_{\theta}\}_{\theta\in R}\subset GL(X)$$T_{y}\phi(x)=\phi(x+y)$,$x\in E^{*}$

It is known that $T_{y}\in \mathcal{L}((E), (E))$ . Moreover,subgroup of $GL((E))$ with infinitesimal generatorofis given as,$\phi\in(E)$is a regular one-parameter. Incidentally, the Fock expansion$\{T_{\theta y}\}_{\theta\in R}$$D_{y}$$T_{y}$$T_{y}= \sum_{n=0}^{\infty}\frac{1}{n!}--(y^{\otimes n})$$Sinoe–(y^{\otimes n})=D_{y}^{n}$.

.

, it follows from Theorem 5.2 that$\phi(x+y)=\sum_{n=0}^{\infty}\frac{1}{n!}(D_{y}\phi)(x)$,136
where the series converges in $(E)$ and therefore pointwisely as well. This is the Taylor
expansion of $\phi\in(E)$ .
EXAMPLE 8.4 (WEYL FORM OF CCR). For $\xi\in E$ we define
$\xi$

(8-4)

$\{_{Q^{\xi}\phi(x)=e^{i(x,\xi\rangle}\phi x)}P_{\xi}\phi(x)=\exp(-\frac{1}{(2}\langle x,$}$- \frac{1}{4}\langle\xi, \xi\rangle)\phi(x+\xi)$,It can be checked that both belong to $GL((E))$ and give rise to unitary representationsof the additive group $E$ . Moreover, put$\{\begin{array}{l}p_{\xi}=\frac{l}{2}(D_{\xi}-D_{\xi}^{*})q_{\xi}=i(D_{\xi}+D_{\xi}^{*})\end{array}$andare regular one-parameter subgroups of $GL((E))$ with
infinitesimal generators and , respectively.
We have introduced white noise coordinate system $\{x(t)\}$ in \S 4. By a similar argument
as in the proof of Lemma 6.2 one can prove easily that
Then,$\{Q_{\theta\xi}\}_{\theta\in R}$$\{P_{\theta\xi}\}_{\theta\in R}$$p_{\xi}$(8-5)$q_{\xi}$$x(t)=\partial_{t}+\partial_{t}^{*}$,$t\in T$,

where $x(t)$ is regarded as multiplication operator. Henoe a white noise analogy of an
infinitesimal generator of finite dimensional rotations is given as(8-6)$x(s)\partial_{t}-x(t)\partial_{S}=(\partial_{s}^{*}+\partial_{S})\partial_{t}-(\partial_{t}^{*}+\partial_{t})\partial_{s}=\partial_{s}^{*}\partial_{\ell}-\partial_{t^{*}}\partial_{s}$.

This is, in fact, an operator in
and we shall investigate its definite meaningin Theorem 8.6 below.
For
we define an operator $d\Gamma(X)$ as follows. Suppose that $\phi\in(E)$ isgiven as$\mathcal{L}((E), (E)^{*})$$X\in \mathcal{L}(E_{C}, E_{\mathbb{C}})$$\phi(x)=\sum_{n=0}^{\infty}\langle:x^{\otimes n}:,$$f_{n}\rangle$,$x\in E^{*}$.

$\gamma_{n}(X)f_{n}\}$,Then we put$d \Gamma(X)\phi(x)=\sum_{n=0}^{\infty}\langle:x^{\otimes n}:,$where$n\geq 1$$\{\gamma_{n}(X)=\sum_{0\gamma o(X)=}n-1I^{\otimes k}\otimes X\otimes I^{\otimes(n-l-k)}$,

137
It is checked easily that $d\Gamma(X)\in \mathcal{L}((E), (E))$ . Formally, $d\Gamma(X)$ is an infinitesimal, whereis a one-parameter subgroup with $X$ being thegenerator of
infinitesimal generator. However, it is not clear whether or not
becomes a
differentiable one-parameter subgroup of $GL((E))$ for any differentiable one-parameter. In this connection regularity introduced above seems
subgroup
useful. In fact, we have the following result.
$\{\Gamma(g_{\theta})\}_{\theta\in R}$$\{g_{\theta}\}_{\theta\in R}$$\{\Gamma(g_{\theta})\}_{\theta\in R}$$\{g_{\theta}\}_{\theta\in R}\subset GL(E)$LEMMA 8.5. If
is a regular one-parameter subgroup of $GL(E)$ with infinitesimal
is a regular one-parameter subgroup of $GL((E))$ withgenemtor $X$ , theninfinitesimal generator $d\Gamma(X)$ .
$\{g_{\theta}\}_{\theta\in R}$$\{\Gamma(g_{\theta})\}_{\theta\in R}$

For the proof we need a long calculation, see [8]. As is easily seen, the infinitesimal
is skew-symmetric in the sense thatgenerator $X$ of$\{g_{\theta}\}_{\theta\in R}\subset O(E;H)$$\langle X\xi, \eta\rangle=-\langle\xi, X\eta\rangle$,$\xi,$$\eta\in E$.

Henoe by a simple argument one comes to the following result including the meaning of
introduced in (8-6).
$x(s)\partial_{t}-x(t)\partial_{s}$THEOREM 8.6 ([8]). Let
be a regular one-parameter subgroup of $O(E;H)$ withis a regular one-parameter subgroup of
infinitesimal genemtor X. Then,$GL((E))$ with infinitesimal genemtor $d\Gamma(X)$ . Moreover, there exists a skew-symmetricsuch that
distribution$\{g_{\theta}\}_{\theta\in R}$$\{\Gamma(g_{\theta})\}_{\theta\in R}$$\kappa\in E\otimes E^{*}$$d \Gamma(X)=\int_{T\cross T}\kappa(s, t)(\partial_{s}^{*}\partial_{t}-\partial_{t^{*}}\partial_{s})dsdt$.

9. Further Topics
Group of Diffeomorphisms

The proof of characterizing the rotation-invariant operators (see \S 7) owes essentiallyto Proposition 7.5. Although we omitted the proof, it requires only a subgeroup of$O(E;H)$ consisting of rotations
such that $ge_{j}=e_{j}$ except finitely many , namely,
which act identically on the subspaoe generated byfor some . Henoe it
is interesting to investigate operators which are invariant under another subgroups of$g$$e_{j}$$\{e_{j}, e_{j+1}, \cdots\}$$j$$O(E;H)$ .

One of the most interesting would be the case of being a (Riemannian) manifoldwith smooth (Riemannian) volume as measure . A diffeomorphism of is calledifadmissible to the Gelfand triple
or to the operator on
$T$

$\nu$

$\gamma$$A$$E\subset L^{2}(T, \nu)\subset E^{*}$$g_{\gamma} \xi(t)=(\frac{d\nu(\gamma^{-1}t)}{d\nu(t)})^{1/2}\xi(\gamma^{-1}t)$,$\xi\in E$,$T$$L^{2}(T, \nu)$138
gives rise to an infinite dimensional rotation $g_{\gamma}\in O(E;H)$ . In other words, is admissible if is stable under . We denote by $Diff_{A}(T)$ the group of admissible diffeomorphisms of . Then it would be very interesting to investigate $Diff_{A}(T)$ -invariant
operators in. The study of $Diff_{A}(T)$ as subgroup of $O(E;H)$ is also deeply
connected with unitary representation theory of a diffeomorphism group, see e.g., [2].
In the special case of $T=R$ with Lebesgue measure and $A=1+t^{2}-d^{2}/dt^{2}$ , we seethat$\gamma$$E$$g_{\gamma}$$T$$\mathcal{L}((E), (E)^{*})$$\int_{R}\partial_{t}dt$

is invariant under $Diff_{A}(T)$ as well as $N$ andalso true.

.
$\Delta_{G}$. We conjecture that the converse isKuo’s Fourier Transform

As in the case of finite dimension “Fourier transform” should be important in harmonic
analysis on Gaussian space. One might think that T-transform introduced in \S 4 would
be one of the candidates of Fourier transform on Gaussian space. However, T-transform
is not a mapping from
into itself and therefore we can not discuss the relation with
operators,
multiplication
opertors, Laplacians and rotations.
differential
Answering a question posed by Hida [3], [4], about a decade ago Kuo invented a
Fourier transform by formal calculus and proved that it intertwines differential operatorsand multiplication operators as usual Fourier transform:$(E)^{*}$$S\partial_{t}=ix(t)\mathfrak{F}$,$Sx(t)=i\partial_{t}S$,in a slightly formal form, see e.g., [6] for a precise statement. There is now a firm groundfor Kuo’s Fourier transform (see [13]) and $S=T^{-1}S$ is one of the equivalent definitionsof Kuo’s Fourier transform, for T- and S-transforms see \S 4. Finally we note that Kuo’s
Fourier transform is the unique (up to a constant factor) continuous linear operatoronwhich possesses the intertwing property mentioned above. The constant isdetermined, for example by, see [6] for details.
$(E)^{*}$$\mathfrak{F}1=\delta_{0}$Volterra Laplacian and L\’evy LaplacianIn the eary years of this century Volterra, G\^ateaux and L\’evy discussed “Laplacians”
acting on functions of infinitely (or rather continuously) many variables, see the book of
L\’evy [15]. Later on various attempts have been made to reformulate their works withmodern language, namely, within the framework of Hilbert spaces or Banach spaces. It
seems also interesting to discuss those operators whithin our setup.
Let $F$ be a C-valued function on $E$ of -class in the sense of Fr\’echet. Sinoe $E$ is$C^{2}$

139
nuclear, for each $\xi\in E$ there exists $F”(\xi)\in(E\otimes E)^{*}$ such that$\frac{d^{2}}{d\theta^{2}}|_{\theta=0}F(\xi+\theta\eta)=\langle F’’(\xi),$$\eta\otimes\eta\rangle$,$\xi,$$\eta\in E$.

ThenF is called an LV-functional if F hasa special form:$\langle F’’(\xi),$$\eta\otimes\zeta\rangle=\int_{T}F_{sing}’’(\xi;t)\eta(t)\zeta(t)dt+\int_{TxT}F_{reg}’’(\xi;s, t)\eta(s)\zeta(t)dsdt$

where
and
and
the regular part of .
$F$
Let be an LV-functional. If the regular part of$H$ , we define$F_{sing}^{\prime l}(\xi;\cdot)\in L_{1^{1}oc}(T)$$F_{reg}^{\prime l}(\xi;\cdot, \cdot)\in L_{1\propto}^{1}(T\cross T)$. We call$F_{sing}’’$,the singular part$F^{n}$$F_{reg}^{n}$$F^{n}$

defines a traoe class operator on$\Delta_{V}F(\xi)=TkaceF_{reg}’’(\xi)=\int_{T}F_{reg}^{u}(\xi;t,t)dt$,

where the integral expression is valid under certain regularity condition. While, if, we put$F_{sing}^{l/}(\xi;\cdot)\in L^{1}(T)$$\Delta_{L}F(\xi)=\int_{T}F_{sing}’’(\xi;t)dt$.

The operators
andare called Volterm Laplacian and L\’evy Laplacian, respectively.
, is a C-valued function on
Recall that the S-transform of $\Phi\in(E)^{*}$ , denoted byand therefore on by restriction. Thus we may discuss the actions ofandon white noise functionals and obtain$\Delta_{V}$$\Delta_{L}$$S\Phi$$E$$E_{C}$$\Delta_{V}S\phi(\xi)=S\Delta_{G}\phi(\xi)$$\Delta_{V}$,$\Delta_{L}S\phi(\xi)=0$,$\xi\in E$,$\phi\in(E)$$\Delta_{L}$.

. While, it isThus we understand that the Volterra Laplacian is an extension of. However, it isfurther proved that the L\’evy Laplacian acts as zero operator on
known thatacts effectively on a spaoe of generalized white noise functionals.
The L\’evy Laplacian is also connected with “asymptotic spherical mean” on Hilbert
spaoe and this justifies the name of Laplacian, see [17]. In their quite recent paper
Accardi, Gibilisco and Volovich [1] investigate a relation between the L\’evy Laplacianand Yang-Mills equations. These works suggest that the L\’evy Laplacian plays a moreinteresting role in infinite dimensional harmonic analysis than we have expected so far.
$\Delta_{G}$$(L^{2})$$\Delta_{L}$140
References
For more complete information on white noise calculus and related topics, see [7],[21], [22] and references cited therein.
1. L. Accardi, P. Gibilisco and I. V. Volovich, The L\’evy Laplacian and the Yang-Mills
equations, Centro V. Volterra preprint series 129 (1992).
2. A. M. Vershik, I. M. Gelfand and M. A. Graev, Representations of the group ofdiffeomorphisms, Russian Math. Surveys 30:6 (1975), 1-50.
3. T. Hida, “Analysis of Brownian Functionals,” Carleton Math. Lect. Notes Vol. 13,
Carleton University, Ottawa, 1975.
4. T. Hida, “Brownian Motion,” Springer-Verlag, 1980.
5. T. Hida, I. Kubo, H. Nomoto and H. Yoshizawa, On projective invariance of Brownian motion, Publ. RIMS, Kyoto Univ. Ser. A 4 (1969), 595-609.
6. T. Hida, H.-H. Kuo and N. Obata, Transformations for white no $ise$ functionals, J.
Funct. Anal. 111 (1993), 259-277.
7. T. Hida, H.-H. Kuo, J. Potthoff and L. Streit, ”White Noise: An Infinite Dimensional
Calculus,” Kluwer Academic, 1993.
8. T. Hida, N. Obata and K. Sait\^o, Infinite dimensional rotations and Lap lacians interms of white noise calculus, Nagoya Math. J. 128 (1992), 65-93.
9. N. K\^ono, Special functions connected with representatio $ns$ of the infinite dimensionalmotion group, J. Math. Kyoto Univ. 6 (1966), 61-83.
10. N. K\^ono, “On Hermite Polynomials,” Seminar on Probability Vol. 27, 1967. (inJapanese)
11. I. Kubo and S. Takenaka, Calculus on Gaussian white noise I-IV, Proc. Japan
Acad. $56A$ (1980), 376-380; 411-416; $57A$ (1981), 433-437; $58A$ (1982), 186-189.
12. I. Kubo and Y. Yokoi, A remark on the space of testing random variables in thewhite no $ise$ calculus, Nagoya Math. J. 115 (1989), 139-149.
13. H.-H. Kuo, Fourier-Mehler tmnsforms in white noise analysis, in “Gaussian Random
Fields (K. It\^o and T. Hida Eds.),” World Scientific, 1991, pp. 257-271.
14. H.-H. Kuo, N. Obata and K. Sait\^o, L\’evy Lap lacian of genemlized functions on a
nuclear space, J. Funct. Anal. 94 (1990), 74-92.
15. P. L\’evy, “Probl\‘emes Concrets d’Analyse Fonctionnelle,” Gauthier-Villars, Paris,

1951.
16. H. Matsushima, K. Okamoto and T. Sakurai, On a certain class of irreducible unitary representations of the infinite dimensional rotation group , Hiroshima Math.
J. 11 (1981), 181-193.
17. N. Obata, The L\’evy Laplacian and the mean value theorem, in “Probability Measureson Groups IX (H. Heyer Ed.),” Lecture Notes in Math. Vol. 1379, Springer-Verlag,
1989, pp. 242-253.
$I$

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18. N. Obata, A characterization of the L\’evy Laplacian in terms of infinite dimensionalrotation gmups, Nagoya Math. J. 118 (1990), 111-132.
19. N. Obata, Rotation-invariant opemtors on white noise functionals, Math. Z. 210(1992), 69-89.
20. N. Obata, An analytic characterization of symbols of opemtors on white noise functionals, J. Math. Soc. Japan 45 (1993), 421-445.
21. N. Obata, Harmonic analysis and infinite dimensional Laplacians on Gaussian
space, Centro V. Volterra preprint series 127 (1992).
22. N. Obata, “Elements of White Noise Calculus,” Lecture Notes in Math., SpringerVerlag, to appear.
23. N. Obata, Opemtor calculus on vector-valued white noise functionals, J. FUnct.
Anal., to appear.
24. K. Okamoto and T. Sakurai, On a certain class of irreducible unitary representations of the infinite dimensional rotation group II, Hiroshima Math. J. 12 (1982),
385-397.
25. K. Okamoto and T. Sakurai, An analogue of Peter- Weyl theorem for the infinite
dimensional unitary gmup, Hiroshima Math. J. 12 (1982), 529-541.
26. A. Orihara, Hermite polynomials and infinite dimensional motion group, J. Math.
Kyoto Univ. 6 (1966), 1-12.
27. Y. Umemura, On the infinite dimensional Laplacian opemtor, J. Math. Kyoto Univ.
4 (1965), 477-492.
28. Y. Umemura and N. K\^ono, Infinite dimensional Laplacian and spherical harmonics,
Publ. RIMS 1 (1966), 163-186.
29. Y. Yamasaki, “Measures on Infinite Dimensional Spaces,” World Scientific, 1985.
30. H. Yoshizawa, Rotation group of Hilbert space and its application to Brownian motion, in “Proc. Internat. Conf. on FMnctional Analysis and Related Topics,” Univ.
of Tokyo Press, 1970, pp. 414-423.
31. H. Yoshizawa, Infinite-dimensional rotation group and Brownian motion, in “Gaussian Random Fields (K. It\^o and T. Hida Eds.),” World Scientific, 1991, pp. 406-419.