数理解析研究所講究録980 巻 1997 年 137-149

137

Generalization of Integral Kernel OperatorsNOBUAKI OBATA
DEPARTMENT OF MATHEMATICSSCHOOL OF SCIENCENAGOYA UNIVERSITYNAGOYA, 464-01 JAPANIntroduction
In most literatures creation and annihilation operators in a Fock space are introduced as
operator-valued distributions though used in actual computation as if they were defined
pointwisely. On the other hand, it is also possible to give a rigorous definition of suchfield operators at a point using a Gelfand triple or a rigged Hilbert space, see e.g., [1], [2].
The so-called white noise calculus initiated by Hida [3] offers one of such possibilities.
The foundation of white noise calculus is a Schwartz type distribution theory on a
; more precisely, it is based on a particular choice of a Gelfand
Gaussian space
triple:$(E)\subset L^{2}(E*, \mu)\subset(E)^{*}$ ,$(E^{*}, \mu)$-Segal isowhere $L^{2}(E*, \mu)$ is isomorphic to a Boson Fock space through themorphism. Then a pointwisely defined annihilation operator, which is also called Hida’s
differential operator and is denoted by , becomes a continuous operator on $(E)$ ; and a.
is a continuous operator on
pointwisely defined creation operator$[10]-[12]$operators on
theorysystematicofawe have established
In a series of works
Gaussian space in terms of white noise calculus. The key role has been played by an
integral kemel operator of which formal integral expression is given as$\mathrm{W}\mathrm{i}\mathrm{e}\mathrm{n}\mathrm{e}\mathrm{r}_{-}\mathrm{I}\mathrm{t}\hat{\mathrm{o}}$$\partial_{t}$$(E)^{*}$$\partial_{t}^{*}$$\int_{T^{\iota+m}}\kappa(_{S\cdots,S}1,l, t1, \cdots,t_{m})\partial*\ldots\partial_{S_{l}}^{*}\partial_{t_{1}}\cdots\partial t_{m}ds_{1}\cdots dS\iota dt_{1}\cdots dt_{m}S1$’(1)

where is a distribution in $l+m$ variables. It should be emphasized strongly that an
is wellintegral kernel can be a distribution. In fact, the composition$(E)$
operatorfromproduct)
becomesacontinuousanddefined (namely, normally ordered. Moreover, the dependence of the parameters and is smooth enough.
into
The kernel distribution in (1) being regarded as a scalar operator-valued distributon,
we are led quite naturally to a generalization with an integral kernel being an operatorvalued distribution. In this note we shall introduce an operator in the following form:$\kappa$$\partial_{s_{1}s}^{*}\ldots\partial^{*}\mathrm{t}\partial t_{1}\ldots\partial_{t_{m}}$$\kappa$$(E)^{*}$$s_{j}$$t_{k}$$\kappa$$\int_{T^{l+m}}\partial_{s_{1}}^{*}\cdots\partial_{s}^{*}L(\iota\ldots, Ss1,l, t_{1}, \cdots, tm)\partial t1\ldots\partial t_{m}ds1\ldots dS\iota dt1\ldots dt_{m}$

.

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Of course, this is a formal (but sometimes very descriptive) expression. For the precise
definition we need the characterization theorem for operator symbols and some propertiesof operator-valued distributions. Those results are obtained in $[10]_{-}[12]$ .
As application we discuss an operator-valued (or quantum) stochastic process of HitsudaSkorokhod type. We shall observe that the classical case discussed in [4] (see also [7], [8])is recovered as multiplication operator-valued processes. Our discussion is closely related
to quantum stochastic calculus, in particular, to representation of quantum martingales,see [6], [9], [13], [14]. Further detailed study in this direction will appear in a forthcoming
paper.
ACKNOWLEDGEMENTS. I am very grateful to Professors I. Kubo and H. Watanabe forinteresting discussion and comments.

1

White noise functionals

We employ the standard setup for white noise calculus $([5], [10]-[12])$ with the same
notation as used there. Let be a topological space with a Borel measure $\nu(dt)=dt$
which is thought of as a time parameter space when it is an interval, or more generally asa field parameter space. Given a positive selfadjoint operator on the real Hilbert spacewith Hilbert-Schmidt inverse, one may form a Gelfand triple:
$T$

$A$

$H=L^{2}(T, \nu;\mathbb{R})$$E\subset H=L^{2}(\tau, \nu;\mathbb{R})\subset E^{*}$

in the standard manner; namely,

$E$

norms

is the$C^{\infty}$-domain of,$A$

equipped with the Hilbertian,whereis the norm of $H$ . Such a countably Hilbert space is called a standard CHspace, see [11]. Since
is of Hilbert-Schmidt type, becomes a nuclear space. The
canonical bilinear form on
and the real inner product of $H$ are denoted by thesame symbol
without contradiction.
One can think of andas spaces of test and generalized functions on , respectively.
In order to keep the delta functions within our discussion we assume:(H1) for each $\xi\in E$ there exists a unique continuous function
such that
$t\in
T$
;
for v-a.e.
$\xi\sim\in E$ , is continuous, i.e.,(H2) for each $t\in T$ a linear functional :$t\inT$(H3) the map, is continuous with respect to the strong dual topology$|\xi|_{\mathrm{p}}=|A^{p}\xi|_{0}$,$\xi\in H$$p\in \mathbb{R}$$|\cdot|_{0}$$A^{-1}$$E$$E^{*}\cross E$$\langle\cdot, \cdot\rangle$$E$$E^{*}$$T$$\delta_{t}$$\xi(t)=\xi(t)\sim$$\xi \mathrm{o}\mathrm{n}T\sim$$\delta_{t}$$\xirightarrow\xi(t),$$\delta_{t}\in E^{*};$$trightarrow\delta_{t}\in E^{*},$$\mathrm{o}\mathrm{f}E^{*}$.

From now on we always assume that every element in
is a continuous function on
and do not use the symbol . For another reason we need one more assumption:(S) inf.
put
We then$E$$\xi\sim$$\mathrm{S}\mathrm{p}\mathrm{e}\mathrm{C}(A)>1$$\delta=||A^{-1}||_{\mathrm{H}\mathrm{S}}<\infty$,$\rho=||A^{-1}||_{\mathrm{o}\mathrm{P}}=(\inf \mathrm{S}\mathrm{p}\mathrm{e}\mathrm{c}(A))^{-1}$.

The obvious inequalities$0<\rho<1$ ;$|\xi|_{\mathrm{P}}\leq\rho^{q}|\xi|_{p+}q$’$\xi\in E$,$p\in \mathbb{R}$,$q\geq 0$

,

$T$

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are used throughout with no special notice.
The Gaussian measureteristic function is:$\mu$

is by definition a probability measure on$\exp(-\frac{1}{2}|\xi|_{0}2)=\int_{E^{*}}e^{i}(x,\epsilon\rangle\mu(dX),$

The probability space$(E^{*}, \mu)$$\xi\in E$

of which charac-$E^{*}$.

is called a Gaussian space. We put$(L^{2})=L^{2}(E^{*}, \mu;\mathbb{C})$for simplicity.
The canonical bilinear form on
is denoted byagain and its
bilinear extension tois also denoted by the same symbol. For $x\in E^{*}$ let:: be defined as a unique element insatisfying$(E^{\otimes n})^{*}\cross(E^{\otimes n})$$\mathbb{C}-$$\langle\cdot, \cdot\rangle$$(E_{\mathbb{C}}^{\otimes n})^{*}\cross(E_{\mathbb{C}}^{\otimes n})$$x^{\otimes n}$$(E^{\otimes n})_{\mathrm{S}}*\mathrm{y}\mathrm{m}$$\phi_{\xi}(X)\equiv\sum_{=n0}^{\infty}\langle:x^{\otimes n}:,$$\frac{\xi^{\otimes n}}{n!}\rangle=\exp(\langle x, \xi\rangle-\frac{1}{2}\langle\xi, \xi\rangle)$

This “normalized” exponential functionis the vacuum. As is well known, each$\xi\in E_{\mathbb{C}}$.

(3)

is called an exponential vector. In particular,
is expressed in the following form:$\phi_{\xi}$$\phi 0$,$\emptyset\in(L^{2})$$fn\otimes n\rangle$$\phi(x)=\sum_{=n0}^{\infty}\langle:X:,$,$x\in E^{*}$,$f_{n}\in H_{\mathbb{C}}^{\otimes n}\wedge$,(4)

where eachand the convergence of the series are understood in the-sense. Expression (4) is referred to as the Wiener-It\^o expansion of . In that case,$xrightarrow$$\langle:x^{\otimes n}:, f_{n}\rangle$$L^{2}$$\phi$(5)$|| \phi||_{0}^{2}\equiv\int_{E^{\mathrm{e}}}|\phi(x)|^{2}\mu(dX)=n=0\sum n!|f_{n}|^{2}\infty 0^{\cdot}$Thus we have a unitary isomorphism between,
and the Boson Fock space overwhich is the celebrated-Segal isomorphism.
, is an operator in
The second quantized operator of , denoted bydefinedby$(L^{2})$$H_{\mathbb{C}}$$\mathrm{W}\mathrm{i}\mathrm{e}\mathrm{n}\mathrm{e}\mathrm{r}-\mathrm{I}\mathrm{t}_{\hat{\mathrm{O}}}$$(L^{2})$$\Gamma(A)$$\mathrm{A}$$\Gamma(A)\phi(x)=n\sum^{\infty}\langle:X^{\otimes}:,$$A^{\otimes}nfn\rangle=0n$,where $\phi\in(L^{2})$ is given as in (4). Equipped with the maximal domain,
becomes-space which will bea positive selfadjoint operator on
and we obtain a standard
denoted by $(E)$ . That
admits a Hilbert-Schmidt inverse is guaranteed by hypothesis(S). Therefore, $(E)$ becomes a nuclear Fr\’echet space and we come to a complex Gelfand
triple:$(E)\subset(L^{2})=L^{2}(E^{*}, \mu;\mathrm{c})\subset(E)^{*}$ .
$\Gamma(A)$$(L^{2})$$\mathrm{C}\mathrm{H}$$\Gamma(A)$Elements in $(E)$ andare called a test (white noise) functional and a generalized(white noise) functional, respectively. We denote by
the canonical bilinear form on$(E)^{*}\cross(E)$ and by, namely,
the norm introduced from$(E)^{*}$$\langle\langle\cdot, \cdot\rangle\rangle$$||\cdot||_{p}$$\Gamma(A)$$|| \emptyset||_{p}^{2}=||\Gamma(A)^{\mathrm{p}}\emptyset||_{0}2=\sum_{n=0}^{\infty}n!|(A^{\otimes n})^{p}fn|_{0}^{2}=\sum^{\infty}n!|n=0f_{n}|_{p}2$

,

(6)

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where andare related as in (4). Thus (5) is a special case of (6). As is easily seenfrom (6),belongs to $(E)$ if and only if
for all and$p\geq0$.
for all
We use a similar (but formal) expression for a generalized white noise functional. Every$(f_{n})_{n=0}^{\infty}$$\phi$$\emptyset\in(L^{2})$$\Phi\in(E)^{*}$$f_{n}\in E_{\mathbb{C}}^{\otimes n}\wedge$

is written as$\Phi(x)=\sum^{\infty}\langle n=0:X^{\otimes}:n,$where$\sum_{n=0}^{\infty}n!|f_{n}|_{p}^{2}<\infty$$n$,$F_{n\rangle}$

(7)

and

$F_{n}\in(E_{\mathbb{C}}^{\otimes n})^{*}\mathrm{s}\mathrm{y}\mathrm{m}$$|| \Phi||_{-p}^{2}=n\sum_{=0}^{\infty}n!|F_{n}|_{-p}^{2}$.

(8)

By construction $||\Phi||_{-p}<\infty$ for some $p\geq 0$ , and hence for all sufficiently large $p\geq 0$ .
Expression (7) is also called the Wiener-It\^o expansion of . In that case,$\Phi$$\langle\langle\Phi, \phi\rangle\rangle=\sum_{n=0}^{\infty}n!\langle F_{n},$

$fn)$

,

where $\phi\in(E)$ and its Wiener-It\^o expansion is given as in (4).

2Integral kernel operators

For any $y\in E^{*}$ and $\phi\in(E)$ we put$D_{y} \phi(x)=\lim_{arrow\theta 0}\frac{\phi(_{X+}\theta y)-\emptyset(_{X})}{\theta}$,$x\in E^{*}$.

(9)

It is known that the limit always exists and that $D_{y}\in \mathcal{L}((E), (E))$ . Since the deltaby hypotheses
functions are elements in, we may define$E^{*}$$\delta_{t}$$(\mathrm{H}1)-(\mathrm{H}3)$$\partial_{t}=D_{\mathit{5}_{t}}$,$t\in T$.

This is called Hida’s differential operator. Obviously,
is a rigorously defined annihilation operator at a point $t\in T$ . It should be therefore emphasized thatis not an
operator-valued distribution but a continuous operator for itself. The creation operator
is by definition the adjoint
and we come to the so-called canonicalcommutation relation:$\partial_{t}$$\partial_{t}$$\partial_{t}^{*}\in \mathcal{L}((E)^{*}, (E)^{*})$$[\partial_{S}, \partial_{t}]=0$,$[\partial_{s}^{*}, \partial_{t}^{*}]=0$,$[\partial_{s}, \partial_{t}^{*}]=\delta_{s}(t)I$,$s,t\in T$

.

(10)

The last relation is understood in a generalized sense.
For $\psi\in(E)$ let
be a function on
defined by$\phi,$$\eta_{\phi,\psi}$$T^{l+m}$$\eta_{\phi,\psi}(s_{1}, \cdots, s_{l}, t_{1}, \cdots, t_{m})=\langle\langle\partial_{s1}^{*}\cdots\partial_{sl}^{*}\partial_{tt}1\ldots\partial\phi,$$\psi m\rangle\rangle$

Then

(11)

and
is a continuous bilinear form on $(E)$ for. By general theory there exists a unique continuous linear operator
such that$\eta_{\phi,\psi}\in E_{\mathbb{C}}^{\otimes(lm}+$$\mathrm{a}\mathrm{n}\mathrm{y}--\kappa\in(E_{\mathbb{C}}^{\otimes(l+m)})^{*}$).

$(\phi, \psi)rightarrow\langle\kappa, \eta_{\phi},\psi\rangle$$rightarrow\iota_{m},(\kappa)\in \mathcal{L}((E), (E)^{*})$$\langle\langle_{-}^{-_{l,m}}-(\kappa)\emptyset, \psi\rangle\rangle=\langle\kappa, \eta_{\phi},\psi\rangle$,$\phi,$$\psi\in(E)$

.

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In other words, $–l,m-(\kappa)$ is defined through two canonical bilinear forms:$\langle\langle_{-}^{-}-_{l,m}(\kappa)\emptyset, \psi\rangle\rangle=\langle\kappa,$$\psi S_{1}s_{\mathrm{t}}1m\rangle\rangle\rangle$$\langle\langle\partial^{*}\cdots\partial^{*}\partial_{t}\cdots\partial_{t}\emptyset,$,$\phi,$$\psi\in(E)$.

This suggests us to employ a formal integral expression:–lm–,$( \kappa)=\int_{T^{l+m}}\kappa(S1, \cdots, S_{l,1}t, \cdots, t_{m})\partial_{s1s}^{*}\ldots\partial^{*}\partial_{t_{1}}\cdots\partial tmdS_{1}\cdots dS_{l}dt\iota 1\ldots dt_{m}$.

an integral kernel operator with kernel distribution . It is noteworthy
and becomes a continuous operator inis defined for any$L((E), (E)^{*})$ . For any $p>0$ with
we have
We callthat$–l.m-(\kappa)$$\kappa$$\kappa\in(E_{\mathbb{C}}^{\otimes(+)}\iota m)^{*}$$–l,m-(\kappa)$$|\kappa|_{-p}<\infty$,$||_{-}^{-}-\prime_{m},(\kappa)\phi||-p\leq Cl,m;p|\kappa|_{-}p||\emptyset||_{p}$$\phi\in(E)$where$c_{l,m;p}=\rho-\mathrm{p}$

1m/2
(llm)

,

(13)

) $/2$$( \frac{\rho^{-p}}{-2pe\log\rho})^{1^{l}}+m$

for all sufficiently large $p>0$ .
This estimate is useful. Recall that
due to relation (10); however, fordetermineduniquelynotisdistribution
The kernelof allthe uniqueness we only need to restrict ourselves to the subspacevariables
which is symmetric with respect to the first and the last
independently.
$|\kappa|_{-p}<\infty$$(E_{\mathbb{C}}^{\otimes(l+}m))_{\mathrm{s}}^{*}\mathrm{y}\mathrm{m}(l,m)$$\kappa\in(E_{\mathbb{C}}^{\otimes(+m)})^{*}l$

3

$m$

$l$

Symbol and Fock expansion
a function on$\mathrm{F}\mathrm{o}\mathrm{r}---\in \mathcal{L}((E), (E)^{*})$$E_{\mathbb{C}}\cross E_{\mathbb{C}}$defined by$—\wedge(\xi, \eta)=\langle\langle_{-}^{-}-\phi_{\xi}, \phi_{\eta}\rangle\rangle$,$\xi,$,$\eta\in E_{\mathbb{C}}$(14)spans a dense sub. Since the exponential vectors
is called the symbol
space of $(E)$ , the synlbol recovers the operator uniquely. For an integral kernel operator,$\{\phi_{\xi;}\xi\in E_{\mathbb{C}}\}$$\mathrm{o}\mathrm{f}---$$–l,m-\overline{(}\kappa)(\xi, \eta)=\langle\kappa,$,$\eta^{\otimes l}\otimes\xi^{\otimes}m\rangle e^{\mathrm{t}\xi,\eta\}}$(15)

or equivalently,$\langle\langle_{-l,m}^{-}-(\kappa)\emptyset\epsilon, \phi_{\eta}\rangle\rangle=\langle\langle\langle\kappa,$$\eta\otimes\otimes l\xi\otimes m\rangle\phi_{\xi},$$\phi_{\eta}\rangle\rangle$$\xi,$$\kappa\in E_{\mathbb{C}}^{\otimes 1^{\iota}+}m$$\eta\in E_{\mathbb{C}}$$\mathcal{L}((E), (E)^{*})$$\xi,$$\xi_{1},$$\eta,$(16)

se.e that)
It is straightforward toand, possesses the following two properties:, the function(O1) For any

where

,

$\eta_{1}\in E_{\mathrm{C}}$$z,$ $wrightarrow\ominus(z\xi+\xi 1, w\eta+\eta_{1})$,is entire holomorphic;(02) There exist constant numbers $C\geq 0,$ $K\geq 0$ and$|\ominus(\xi, \eta)|\leq C\exp I\mathrm{t}’(|\xi|_{p}^{2}+|\eta|_{p}2)$$z,$$w\in \mathbb{C}$$p\in \mathbb{R}$,,such that$\xi,$$\eta\in E_{\mathbb{C}}$.

$\ominus=---\wedge,$$—\in$

142

More important is that the converse is also true.
Theorem $bl3.1$ Anyis the symbol of an$\mathbb{C}$-valued function

on$\Theta$$E_{\mathbb{C}}\cross E\mathrm{c}_{\underline{\wedge}}satisfyin=\ominus g$$operat_{\mathit{0}}r---\in \mathcal{L}((E), (E)^{*}),$$i.e.$, –conditions $(Ol)$ and$(O\mathit{2})$, there exists a unique family of kernel distributions$\kappa_{l,m}\in(E_{\mathbb{C}}^{\otimes}m)_{\mathrm{S}}*\mathrm{C}(\iota,m)\mathrm{S}\mathrm{u}\mathrm{h}\mathrm{t}\mathrm{I}\mathrm{n}\mathrm{f}\mathrm{a}\mathrm{C}\mathrm{t},\ovalbox{\tt\small REJECT}_{+}^{\mathrm{i}\mathrm{v}}\mathrm{e})\mathrm{n}\mathrm{S}\mathrm{u}\mathrm{c}\mathrm{y}\mathrm{m}\mathrm{h}\mathrm{a}\mathrm{f}\mathrm{u}\mathrm{n}\mathrm{C}\mathrm{t}\mathrm{i}\mathrm{h}\mathrm{a}\mathrm{t}\mathrm{o}\mathrm{n}\ominus$$\ominus.(\xi, \eta)=,\sum_{0\iota_{m=}}^{\infty}\langle\langle---l,m(\kappa_{l,m})\phi\epsilon, \phi_{\eta}\rangle\rangle$,$\xi,$$\eta\in E_{\mathbb{C}}$.

Moreover, the series$– \phi-=l,\sum_{m=0}^{\infty}---_{\iota_{m}},(\kappa_{l,m})\emptyset$

converges in,$\phi\in(E)$

,

(17)

, and thereby we obtain $—\in \mathcal{L}((E), (E)^{*})$ of which symbol is . In
particular, the symbol
satisfying (O1) and (02), the above argument
reproduces anterms of integral kernel operators. Expression (17) is called
the Fock expansion.
In some practical problems operators on Fock space are only defined on the exponential
vectors
due to the fact that they are linearly independent. Theorem 3.1 istherefore crucial for checking whether the operator comes into our framework. In fact,
our later discussion will depend on this point heavily. For detailed proof and further
discussion see [10]. Here we do not mention anything about the case ofwhich
is also important from some applications. For complete information see [11].
$(E)^{*}$$O-$$\mathrm{o}\mathrm{f}---\in \mathcal{L}((E), (E)^{*})$$\mathrm{o}\mathrm{p}\mathrm{e}\mathrm{r}\mathrm{a}\mathrm{t}\mathrm{o}\mathrm{r}---\mathrm{i}\mathrm{n}$$\mathrm{o}\mathrm{f}---$$\{\emptyset\epsilon;\xi\in E_{\mathbb{C}}\}$$\mathcal{L}((E), (E))$

4Operator-valued distributions

In [12] we studied-valued distributions in general, where is a standard CHspace. Here we recapitulate some results for.
Let
be the normalized eigenfunctions of the operator . Forand
we put$\mathcal{L}(\mathcal{E}, \mathcal{E}^{*})$$\mathcal{E}$$\mathcal{E}=(E)$$A$$\{e_{j}\}_{j=}^{\infty}0$$\mathrm{i}=(\dot{i}_{1}, \cdots, i_{l})$$\mathrm{j}=(j_{1}, \cdots,j_{m})$$e(\mathrm{i})=e_{i_{1}}\otimes\cdots\otimes e_{i}l$

For a linear map’$L:E_{\mathbb{C}}^{\otimes}(\iota+m)arrow \mathcal{L}((E), (E)^{*})$$e(\mathrm{j})=e_{j_{1}}\otimes\cdots\otimes e_{j_{m}}$and$p,$ $q,$ $r,$ $s\in \mathbb{R}$

we put$||L|| \iota_{mp,q;},;r,s=\sup\dagger\sum_{\mathrm{i}_{\dot{\mathrm{d}}}}|\langle\langle L(e(\mathrm{i})\otimes e(\mathrm{j}))\emptyset, \psi)\rangle|2|e(\mathrm{i})|_{p}^{2}|e(\mathrm{j})|^{2}q$

By definition for any$p,$ $q,$ $r,$ $s\in \mathbb{R}$.

;$\phi,\psi\in(||\emptyset|||\psi|^{1}|-r\leq 1-s\leq 1E)\}^{1/2}$

we have$\sum_{\mathrm{i}_{\dot{\mathrm{d}}}}|\langle\langle L(e(\mathrm{i})\otimes e(\mathrm{j}))\emptyset, \psi\rangle\rangle|^{2}|e(\mathrm{i})|^{2}p|e(\mathrm{j})|_{q}^{2}\leq||L||^{2}\iota_{m;},p,q;r,s||\phi||2-S||\psi||_{-}^{2}r$

(18)

and
$||L||_{l,r}m;p,q;,S\leq\rho^{\iota}|p+\prime mqJ|L||_{l,q’}m;p+p’,q+;r+rS+’,S’$

’

$p’,$

$q”,$
$r,$ $S’\geq 0$ .

(19)

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For brevity we put$||L||_{l,;}mp,q=||L||_{l,;p,q}m;p,q$$||L||_{p}=||L||_{l,p}m;p,p;p,$ ’.

The next result will be useful, for the proof see [12].
Proposition 4.1 For a linear map
tio $ns$ are equivalent:;(i)$L:E_{\mathbb{C}}^{\otimes \mathrm{t}^{l}+m)}arrow \mathcal{L}((E), (E)^{*})$

the $foll_{ow}ingfour$ condi-$L\in \mathcal{L}(E_{\mathbb{C}}^{\otimes}m)\mathcal{L}\{l+,((E), (E)*))$(ii)$\sup\{|\langle\langle L(\eta)\phi, \psi\rangle\rangle|;\eta\in E_{\mathbb{C}}\phi,\psi\in\otimes(E^{+})(lm,),$$|1|\psi||_{p}\leq 1\}<\infty$$||\phi|||\eta|_{p}\leq \mathrm{P}\leq 1,$(iii) $||L||_{-p}<\infty$ for some(iv) $||L||_{l,m;p},q;r,S<\infty$ for some
we have
In that case, for any

for some$p\geq 0_{j}$$p\geq 0_{i}$$p,$ $q,$ $r,$$p,$ $q,$ $r,$$s\in \mathbb{R}$.

$s\in \mathbb{R}$$|\langle\langle L(\eta)\emptyset, \psi\rangle\rangle|\leq||L||_{l,r}m;-p,-q;-,-S|\eta|l,m;p,q||\phi||S||\psi||_{r}$

,

(20)

and
$||L(\eta)\emptyset||-r\leq||L||l,m;-p,-q;-r,-s|\eta|_{l,q}m;p,||\phi||S$

where$\eta\in E_{\mathbb{C}}^{\otimes(+\rangle}\iota m,$

Eachbution onmorphism$\phi,$$\psi\in(E)$

. If

(21)

.
-valued distriis justifiably called an
were a Fr\’echet space, one would have a canonical iso-$L\in \mathcal{L}(E_{\mathbb{C}}^{\otimes(m)}+, \mathcal{L}(l(E), (E)^{*}))$$T^{l+m}$,$\mathcal{L}((E), (E)^{*})$$\mathcal{L}((E), (E)^{*})$$\mathcal{L}(E_{\mathbb{C}}^{\otimes(m)}, \mathcal{L}+((\iota E), (E)^{*}))\cong(E_{\mathbb{C}}^{\otimes(+}\iota m))*\otimes \mathcal{L}((E), (E)^{*})$

by the kernel theorem; however,is Fr\’echet if and only if$\mathcal{L}(\mathcal{E}, \mathcal{E}^{*})$

5$\mathcal{L}((E), (E)^{*})$$\mathcal{E}$is not a Fr\’echet space. (It is known thatis a Hilbert space.)

Generalization of integral kernel operators

With eachthe formula:$L\in \mathcal{L}(E_{\mathbb{C}}^{\otimes\langle m)}l+, \mathcal{L}((E), (E)^{*}))$

we associate an operator$\langle(_{-}^{-}-\phi_{\xi}, \phi_{\eta}\rangle\rangle=\langle\langle L(\eta^{\otimes l}\otimes\xi\otimes m)\phi\epsilon,$$\phi_{\eta}\rangle\rangle,$$\xi,$$—\in \mathcal{L}((E), (E)^{*})$$\eta\in E_{\mathbb{C}}$

.

by

(22)

We must check that the definition works; namely, conditions (O1) and (02) in \S 3 are to
be verified for$t$$\ominus(\xi, \eta)=\langle\langle L(\eta^{\otimes l}\otimes\xi\otimes m)\phi\epsilon,$$\phi_{\eta}\rangle\rangle$,$\xi,$$\eta\in E_{\mathbb{C}}$

.

(23)

In fact, the verification of (O1) is straightforward. As for (O2), it follows from (20) in
Proposition 4.1 that$|\ominus(\xi, \eta)|$$\leq$$=$$||L||_{-\mathrm{p}}|\eta^{\otimes l}\otimes\xi^{\otimes}m|_{p}||\phi_{\xi}||_{p}||\phi\eta||_{p}$$||L||-p| \eta|_{p}^{l}|\xi|^{m}p\exp\frac{1}{2}(|\xi|p+|\eta 2|^{2}p)$

.

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Hence one may find $p\geq 0,$ $K\geq 0$ and $C\geq 0$ such that$|\ominus(\xi, \eta)|\leq C\exp I\iota’(|\xi|_{\mathrm{P}^{+}}^{2}|\eta|_{p}^{2})$,$\xi,$$\eta\in E_{\mathbb{C}}$,which shows (02). It then follows from the characterization theorem (Theorem 3.1) that; namely, there exists a unique operator
is the symbol of ansatisfying (22). It is reasonable to write$\ominus$$\mathrm{o}\mathrm{p}\mathrm{e}\mathrm{r}\mathrm{a}\mathrm{t}\mathrm{o}\mathrm{r}---\in \mathcal{L}((E), (E)^{*})$$—\in \mathcal{L}((E), (E)^{*})$

.

(24)

$—= \int_{T}\partial_{s}^{*}1\ldots\partial_{s\iota}*L(s1, \cdots, s\iota, t_{1}, \cdots, tm)\partial_{t_{1}}\cdots\partial tmd_{S_{1}}\cdots dS_{l}dt1\ldots dt_{m}$The above constructeda generalization of an integral kernel operator introduced in \S 2, compare (22) and (16).
This genera,lization occurs in an integral kernel operator. Let
and. To go further we need contraction of tensor
consider an integral kernel
products. For
and
we definea uniqueelement satisfying$\mathrm{o}\mathrm{p}\mathrm{e}\mathrm{r}\mathrm{a}\mathrm{t}\mathrm{o}\mathrm{r}---\mathrm{i}\mathrm{s}$$\kappa\in(E_{\mathbb{C}}m))_{\mathrm{s}\mathrm{y}}^{*}\otimes(l+\mathrm{m}(l,m)$$\mathrm{o}\mathrm{p}\mathrm{e}\mathrm{r}\mathrm{a}\mathrm{t}_{\mathrm{o}\mathrm{r}}--_{l,m}-(\kappa)$$g_{l}\in E_{\mathbb{C}}^{\otimes l}$$\kappa\otimes^{l}(g_{l}\otimes g_{n})\in(E_{\mathbb{C}}^{\otimes(m+n)})*\mathrm{a}\mathrm{s}$$g_{n}\in E_{\mathbb{C}}^{\otimes n}$$\langle\kappa\otimes^{l}(gl\otimes gn), \zeta\rangle=\langle\kappa\otimes g_{n}, g_{l}\otimes\zeta\rangle$

Then
is defined for any
Moreover, it is easily verified that$g\in E_{\mathbb{C}}^{\otimes(l)}+m$$\kappa\otimes^{l}g$$|F\otimes^{\iota_{g}}|_{-p}\leq\rho^{2pn}|F|-p|g|p$

The right contraction$\kappa\otimes_{l}g$Lemma 5.1 Fix integers,$\zeta\in E_{\mathbb{C}}^{\otimes()}m+n$.

by continuity and is called a left contraction.
$F\in(E_{\mathbb{C}}^{\otimes(+})\iota m)*$’,$g\in E_{\mathbb{C}}^{\otimes(+n)}l$

.

(25)

is similar. For detailed argument see [11].

$0\leq\alpha\leq l$

and $0\leq\beta\leq m$ . Given$\kappa\in(E_{\mathbb{C}}^{\otimes(}l+m))_{\mathrm{s}\mathrm{y}}^{*}\mathrm{m}(l,m))$$L_{0}(\eta 1, \cdots, \eta\alpha’\xi 1, \cdots, \xi\beta)=--l_{-}\alpha,m-\beta(-(\kappa\otimes\beta(\xi 1\otimes\cdots\otimes\xi_{\beta}))\otimes\alpha(\eta_{1}\otimes\cdots\otimes\eta_{\alpha}))$becomes a continuous$(\alpha+\beta)$-linear map from

$E_{\mathbb{C}}$into$\mathcal{L}((E), (E)^{*})$.

PROOF. For simplicity we put.

$\lambda=(\kappa\otimes_{\beta(\xi}1\otimes\cdots\otimes\xi\beta))\otimes^{\alpha}(\eta 1\otimes\cdots\otimes\eta_{\alpha})$Take $p>0$ with$|\kappa|_{-\rho}<\infty$. Then by (25) we have$|\lambda|_{-p}\leq|\kappa|_{-p}|\xi_{1}|_{p}\cdots|\xi\beta||\eta 1|p\ldots|\mathrm{P}|_{p}\eta_{\alpha}$

.

(26)

On the other hand, in view of (13) we have$|\langle\langle L_{0}(\eta_{1}, \cdots, \eta\alpha’\xi_{1}, \cdots, \xi\beta)\emptyset, \psi\rangle\rangle|\leq c_{l_{-}\alpha},m-\beta,p|\lambda|_{-}p||\emptyset||_{p}||\psi||p$Given bounded subsets$\sup$$B_{1},$$B_{2}\subset(E)$

.

(27)

, we see from (26) and (27) that$|\langle\langle L_{0}(\eta_{1}, \cdots,\eta\alpha’\xi_{1}, \cdots, \xi\beta)\emptyset, \psi\rangle\rangle|$$\emptyset\in B_{1,\psi\in B_{2}}$$\leq c_{\iota-\alpha,m}-\beta,p\sup_{\emptyset\in B_{1}}||\phi||_{p}\sup_{\psi\in B2}||\psi||p|\kappa|_{-p}|\xi_{1}|_{p}\cdots|\xi_{\beta}|_{p}|\eta 1|p\ldots|\eta_{\alpha}|_{p}$

,

145

which implies the desired continuity of

By Lemma 5.1 there exists$L_{0}$.

qed$L\in \mathcal{L}(E_{\mathbb{C}}^{\otimes\{\beta)}\alpha+, \mathcal{L}((E), (E)^{*}))$

such that.

$L(\eta_{1^{\otimes}}\cdots\otimes\eta\alpha\otimes\xi_{1^{\otimes\cdots\otimes\xi}}\beta)=L0(\eta 1, \cdots, \eta\alpha’\xi 1, \cdot\cdot, ,\xi\beta)$

In other words,$L(\eta_{1}\otimes\cdots\otimes\eta\alpha\otimes\xi 1\otimes\cdots\otimes\xi\rho)$$=_{-l_{-}\alpha,m}--(-\beta(\kappa\otimes_{\beta}(\xi 1\otimes\cdots\otimes\xi_{\beta}))\otimes\alpha(\eta_{1}\otimes\cdots\otimes\eta_{\alpha}))$

.

(28)

Theorem 5.2 (FUBINI TYPE) Fix integers $0\leq\alpha\leq l$ and $0\leq\beta\leq m$ . Givenbe defined as in (28). Then,let$(E_{\mathbb{C}}^{\otimes \mathrm{t}l+m)})^{*}$$L\in \mathcal{L}(E_{\mathbb{C}}^{\otimes \mathrm{t}}\alpha+\beta),$$\mathcal{L}((E), (E)^{*}))$$–l,m-( \kappa)=\int_{T^{\alpha+\beta}}\partial*\ldots\partial_{S_{q}}^{*}L(_{S_{1,\alpha}}s_{1}\ldots,)s,t_{1}, \cdot*\cdot, t_{\beta}\partial_{t_{1}t}\ldots\partial ds_{1}\cdots dS\beta\alpha dt_{1}\cdots dt_{\beta}$

PROOF. The symbol of the right hand side is
In view of (28) we obtain$\langle\langle L(\eta^{\otimes\otimes}\otimes\alpha\xi\beta)\phi\epsilon,$$\kappa\in$$\phi_{\eta}\rangle\rangle$$\langle\langle L(\eta^{\otimes\alpha}\otimes\xi^{\otimes\beta})\phi\xi, \phi_{\eta}\rangle\rangle$

by definition (22).

$=$$\langle\langle_{-l-}^{-}-\alpha,m-\rho((\kappa\otimes\beta\xi^{\otimes\beta})\otimes^{\alpha\otimes}\eta)\alpha\phi\xi,$$=$$\eta-\alpha)\otimes\alpha\otimes \mathrm{t}l\otimes\xi\otimes(m-\beta)\rangle e\mathrm{t}^{\xi,\eta)}$$\langle(\kappa\otimes_{\beta\xi^{\otimes\beta}})\otimes^{\alpha}\eta,$$=$$\langle\kappa,$$\eta\xi\otimes l_{\otimes}\otimes m\rangle e^{(}\xi,\eta\rangle$.

$\phi\eta\rangle\rangle$.

by (15) and hence follows the
The last expression coincides with the symbol
qed
assertion.
The above result is essential to discuss “canonical form” of an adapted operator-valued
process. This topic will be discussed in a forthcoming paper.
$\mathrm{o}\mathrm{f}_{-l,m}--(\kappa)$

6

Operator-valued Hitsuda-Skorokhod integrals

In this section we take
$T=\mathbb{R}$,$A=1+t^{2}- \frac{d^{2}}{dt^{2}}$,$E=S(\mathbb{R})$.

According to the discussion in the previous section we have a generalized integral kernel
operator:$\int_{T}\partial_{l^{*}}L(t)dt$,$L\in \mathcal{L}(E_{\mathbb{C}}, \mathcal{L}((E), (E)^{*}))$.

In this section we shall introduce a “stochastic integral” of the form:$\int_{0}^{t}\partial_{s}^{*}L(s)dS$,$t\geq 0$.

For that purpose should possess a stronger property that is continuously extended. Note the natural inclusion relation
to a linear map from
into
$L$

$L$

$E_{\mathbb{C}}^{*}$$\mathcal{L}((E), (E)^{*})$$\mathcal{L}(E_{\mathbb{C}}^{*}, \mathcal{L}((E), (E)^{*}))\subset \mathcal{L}(E_{\mathbb{C}}, \mathcal{L}((E), (E)^{*}))$.

we write $L_{s}=L(\delta_{s})$ for simplicity. Then
operator-valued(orquantum) stochastic process with values in
as an
For

$L\in \mathcal{L}(E_{\mathbb{C}}^{*}, \mathcal{L}((E), (E)^{*}))$$\{L_{s}\}$is regarded$\mathcal{L}((E), (E)^{*})$

.

146

Lemma 6.1 Let$L\in \mathcal{L}(E_{\mathbb{C}}^{*}, \mathcal{L}((E), (E)^{*}))$

such that$M_{f}\in \mathcal{L}((E), (E)^{*})$. Then for any$\langle\langle M_{f}\phi C, \phi_{\eta}\rangle\rangle=\langle\langle L(f\eta)\emptyset\epsilon, \phi_{\eta}\rangle\rangle$

Moreover,$frightarrow M_{f}$is continuous, $i.e.,$,there exists an operator$f\in E_{\mathbb{C}}^{*}$$\xi,$$\eta\in E_{\mathbb{C}}$$M\in \mathcal{L}(E_{\mathbb{C}}^{*}, \mathcal{L}((E), (E)^{*}))$.

.

PROOF. First note that, for any $p\geq 0$ there exist $q>0$ and $A_{p,q}\geq 0$ such that$|\xi\eta|_{p}\leq A_{\rho,q}|\xi|p+q|\eta|_{p+q}$,$\xi,$$\eta\in E_{\mathbb{C}}$.

Then, by duality we obtain$|f\eta|_{-}(p+q)\leq A_{\rho,q}|f|-p|\eta|_{p+q}$,$\eta\in E_{\mathbb{C}}$,$f\in E_{\mathbb{C}}^{*}$

.

(29)

On the other hand, using the canonical isomorphism$\mathcal{L}(E_{\mathbb{C}}^{*}, \mathcal{L}((E), (E)^{*}))\cong \mathcal{L}(E_{\mathbb{C}}^{*}, ((E)\otimes(E))^{*})$

which comes from the kernel theorem, we find,such that$L^{*}\in \mathcal{L}((E)\otimes(E), E_{\mathbb{C}})$$(\langle L(f)\emptyset, \psi\rangle\rangle=(f, L^{*}(\phi\otimes\psi)\rangle,$

By continuity, for any $p\geq 0$ there exist $q\geq 0$ and$f\in E_{\mathbb{C}}^{*}$,$B_{p,q}\geq 0$,$|L^{*}(\emptyset\otimes\psi)|p\leq B_{p},q||\phi||_{\rho+}q||\psi||_{p+q}$$\phi,$$\psi\in(E)$.

such that$\phi,$$\psi\in(E)$

.

(30)

We now consider.

$\ominus_{f}(\xi, \eta)=\langle\langle L(f\eta)\phi_{\xi}, \phi_{\eta}\rangle\rangle=\langle f\eta, L^{*}(\phi_{\epsilon}\otimes\phi_{\eta})\rangle$

Suppose $p\geq 0$ is given albitrarily. Take $q>0$ with property (29). In view of (30) wemay find $r\geq 0$ such that$|\ominus_{f}(\xi, \eta)|$$\leq$$|f\eta|_{-}(p+q)|L*(\emptyset\xi^{\otimes}\emptyset_{\eta})|_{p+}q$$\leq$$A_{p,q}|f|-p|\eta|p+qB+q,r|p|\phi\xi||_{p+}q+r||\phi_{\eta}||p+q+r$$\leq$$A_{p,q}B_{\rho+}r \rho^{r}q,|f|-p|\eta|_{p+q}+\Gamma\exp\frac{1}{2}(|\xi|_{p++r}^{2}q+|\eta|_{p+q+}2)r$.

Consequently, for any $p\geq 0$ we have found constants $C\geq 0,$ $K\geq 0$ and $s\geq 0$ such that$|\ominus_{f}(\xi, \eta)|\leq C|f|_{-p}\exp Ic(|\xi|_{p+S^{+}}^{2}|\eta|_{p+S}2)$,$f\in E_{\mathbb{C}}^{*}$

Hence by the characterization theorem (Theorem 3.1), for any
operatorsuch that,$\xi,$$\eta\in E_{\mathbb{C}}$$f\in E_{\mathbb{C}}^{*}$

.

(31)

there exists an$\Lambda/I_{j}\in \mathcal{L}((E), (E)^{*})$$\langle\langle M_{f}\phi_{\xi}, \phi_{\eta}\rangle\rangle=\ominus_{f}(\xi, \eta)=\langle\langle L(f\eta)\phi\epsilon, \phi_{\eta}\rangle\rangle$

Obviously,$f\vdasharrow \mathbb{J}l_{f}$$\xi,$$\eta\in E_{\mathbb{C}}$.

is linear. Inequality (31) implies the continuity on the Hilbert space. Sinceis the inductive limit of such Hilbert spaces, we conclude.
qed$\{f\in E_{\mathbb{C}}^{*}; |f|_{-p}<\infty\}$

that

,

$E_{\mathbb{C}}^{*}$$M\in \mathcal{L}(E_{\mathbb{C}}^{*}, \mathcal{L}((E), (E)^{*}))$

147

The operator$M_{f}$

constructed above is denoted by$M_{f}= \int_{T}f(s)\partial*L_{s}dss$.

In particular, for $f=1_{1^{0,t}1}$ we write$\Omega_{t}\equiv\int_{0}^{t}\partial_{s}^{*}L_{s}d_{S}$,$t\geq 0$,. This is called anwhich forms a one-parameter family of operators in
rephrase the definiwe
To
be
sure
type.
operator-valued integral of Hitsuda-Skorokhod
tion:
(32)
.
,
$\mathcal{L}((E), (E)^{*})$$\xi,$$\langle\langle\Omega_{t}\phi\epsilon, \phi_{\eta}\rangle\rangle=\langle\langle L(1[0,t]\eta)\phi\epsilon, \phi_{\eta})\rangle$$\eta\in E_{\mathbb{C}}$

Hitsudageneralizes
It is interesting to observe how our operator-valued process
Skorokhod integral.
For that purpose we quote the definition of the Hitsuda-Skorokhod integral following$t\geq 0$ , be given. Since
for any , for any $\phi\in(E)$[4]. Let. Assume that the function is measurable andone obtains a function:$\mathrm{t}_{\mathrm{J}}\mathrm{h}\mathrm{e}$$\{\Omega_{t}\}$$t$$\partial_{t}^{*}\in \mathcal{L}((E)^{*}, (E)^{*})$$\Phi_{t}\in(E)^{*},$$trightarrow\langle\langle\partial_{t}^{*}\Phi_{t}, \phi\rangle\rangle$$\int_{0}^{t}|\langle\langle\partial^{*}\Phi ss’\phi\rangle\rangle|ds<\infty$

Then, it is proved that there exists$\Psi_{t}$$t\geq 0$$t\geq 0$$\Psi_{t}\in(E)^{*},$$\langle\langle\Psi_{t}, \phi\rangle\rangle=\int_{0}^{t}\langle\langle\partial_{s}*\Phi_{s}, \phi\rangle\rangle ds$

The above obtained,.

, uniquely such that,$\phi\in(E)$.

is denoted by$\Psi_{t}=\int_{0}^{t}\partial^{*}S\Phi_{s}dS$

and is called the Hitsuda-Skorokhod integral. As is well known, the Hitsuda-Skorokhod
is an adapted
integral coincides with the usual It\^o integral when the integrandmotionBrownianbythefunction with respect to the filtration generated$\{\Phi_{t}\}$$B_{t}(x)=\langle x,$$1_{[0,t]}\rangle$,$x\in E^{*}$,$t\geq 0$$L^{2_{-}}$.

In this connection see also [7], [8].
We need one more remark. Each $\Phi\in(E)^{*}$ gives rise to a continuous operator in
is a continuous bilinear map fromby lnultiplication since$(E)$ into $(E)$ . This identification extends to a natural inclusion relation$(E)\cross$$(\phi, \psi)rightarrow\phi\psi$$\mathcal{L}((E), (E)^{*})$$\mathcal{L}(E_{\mathbb{C}}^{*}, (E)^{*})\subset$$\mathcal{L}(E_{\mathbb{C}}^{*}, \mathcal{L}((E), (E)^{*}))$.

Now suppose we are given
cation operator, i.e.,of Hitsuda-Skorokhod type:. Let denote the corresponding multipli. Then one has an operator-valued integral$\tilde{\Phi}$$\Phi\in \mathcal{L}(E_{\mathbb{C}}^{*}, (E)^{*})$$\tilde{\Phi}\in \mathcal{L}(E_{\mathbb{C}}^{*}, \mathcal{L}((E), (E)^{*}))$$\Omega_{t}=\int_{0}^{t}\partial_{S}*\tilde{\Phi}_{s}d_{S}$,$t\geq 0$

,

(33)

148

as well as the Hitsuda-Skorokhod integral in the original sense:$\Psi_{t}=\int_{0}^{t}\partial_{s}*\Phi_{s}d_{S}$,$t\geq 0$

In fact, since the both maps
and
. Therefore is well defined.
$trightarrow\delta_{t}\in E^{*}$$trightarrow\langle\langle\partial_{t}^{*}\Phi_{t}, \emptyset\rangle\rangle$$\Phi\in \mathcal{L}(E_{\mathbb{C}}^{*}, (E)^{*})$$\Psi_{t}=\Omega_{t}\phi 0$$\phi_{0}$(34)$trightarrow\partial_{t}\phi\in(E)$are continuous, so is$\Psi_{t}$Theorem 6.2 For any
let
Skorokhod type defined as in (33) and let
original sense defined as in (34). Then,

where

.

$\Omega_{t}$$\Psi$

be the operator-valued integral of Hitsudabe the Hitsuda-Skorokhod integral in the,$t\geq 0$,

is the vacuum.

PROOF. By definition (32) we have$\langle\langle\Omega_{t}\emptyset 0, \phi\eta\rangle\rangle=\langle\langle\tilde{\Phi}(1_{[0,t}]\eta)\phi 0,$

In terms of the adjoint operator$\langle\langle\Phi(1[0,t]\eta),$$\emptyset\eta\rangle\rangle=\langle\langle\Phi(1_{[}0,t]\eta),$$\Phi^{*}\in \mathcal{L}((E), E_{\mathbb{C}})$$\emptyset\eta\rangle\rangle=\langle 1_{[t]\eta}0,,$$\phi_{\eta\rangle\rangle}$.

the last expression becomes.

$\Phi^{*}\phi\eta\rangle=\int_{0}^{t}\eta(s)(\Phi*\phi_{\eta})(S)d_{S}$Moreover, note that$\uparrow \mathfrak{j}(S)(\Phi*\phi_{\eta})(s)$$=\eta(s)\langle\delta_{S}, \Phi^{*}\phi\eta\rangle=\eta(s)\langle\langle\Phi(\delta S), \phi_{\eta}\rangle\rangle$$=$$\langle\langle\Phi(\delta_{S}), \partial s\phi\eta\rangle\rangle=\langle\langle\partial_{S}*\Phi\psi s’\eta\rangle\rangle$.

Consequently,$\langle\langle\Omega_{t}\phi 0, \phi\eta\rangle\rangle=\int_{0}^{t}\langle\langle\partial_{s}^{*}\Phi_{s}, \phi\eta\rangle\rangle ds=\langle\langle\Psi_{\iota}, \phi_{\eta}\rangle\rangle$

and we come to$\Omega_{t}\phi_{0}=\Psi_{t}$,as desired.

qed

References[1] N. N. Bogolubov, A. A. Logunov and I. T. Todorov: “Introduction to Axiomatic
Quantum Field Theory,” Benjamin, 1975.

[2] A. Grossman: Fields at a point, Commun. Math. Phys. 4 (1967), 203-216.

[3] T. Hida: “Analysis of Brownian Functionals,” Carleton Math. Lect. Notes, no. 13,
Carleton University, Ottawa, 1975.
[4] T. Hida, H.-H. Kuo, J. Potthoff and L. Streit: “White Noise,” Kluwer Academic,

1993.

149

[5] T. Hida, N. Obata and K. Sait\^o: Infinite dimensional rotations and Laplacians interms of white noise calculus, Nagoya Math. J. 128 (1992), 65-93.
[6] R. L. Hudson and K. R. Parthasaraty: Quantum Ito’s formula and stochastic evolutions, Comlnun. Math. Phys. 93 (1984), 301-323.

Browniarifunctionals, in “Theory and Applica[7] I. Kubo: Ito formula for generalized
tion of Random Fields (G. Kallianpur ed.),” pp. 156-166, Lect. Notes in Math. Vol.
49, Springer-Verlag, 1983.
[8] I. Kubo and S. Takenaka: Calculus on Gaussian white noise I-IV, Proc. Japan Acad.
(1982),
411-416; 57A (1981), 433-437;
56A (1980),

376.-$380_{;}$

58A.

[9] P. A. Meyer: “Quantum Probability for Probabilists,” Lect.
1538, Springer-Verlag, 1993.

186-18..9.

$\mathrm{N}\mathrm{o}\mathrm{t}\mathrm{e}\mathrm{s}\wedge$in Math. Vol.

[10] N. Obata: An analytic characterization of symbols of operators on white noise functionals, J. Math. Soc. Japan 45 (1993), 421-445.

[11] N. Obata: “Elements of White Noise Calculus,” Lect. Notes in Math., SpringerVerlag, to be published.
[12] N. Obata: Operator calculus on vector-valued white noise functionals, to appear in
J. Funct. AnaJ. 120 (1994).
[13] K. R. Parthasarathy: “An Introduction to Quantum Stochastic Calculus,” Birk-h\"auser, 1993.
[14] K. R. Parthasarathy and K. B. Sinha: Stochastic integral representation of bounded
quantum martingales in Fock space, J. Funct. Anal. 67 (1986), 126-151.