Generalization of Integral Kernel Operators

Generalization

of

Integral Kernel

Operators

NOBUAKI OBATA DEPARTMENT OF MATHEMATICS

SCHOOL

OF SCIENCE NAGOYA UNIVERSITY NAGOYA,

464-01

JAPAN

Introduction

In most literatures creation and annihilationoperators 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 such

fieldoperators 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

Gaussian space $(E^{*}, \mu)$;

more

precisely, it is based

on a

particular choice of

a

Gelfand

triple:

$(E)\subset L^{2}(E*, \mu)\subset(E)^{*}$,

where $L^{2}(E*, \mu)$ is isomorphic to

a

Boson Fock space through the $\mathrm{W}\mathrm{i}\mathrm{e}\mathrm{n}\mathrm{e}\mathrm{r}_{-}\mathrm{I}\mathrm{t}\hat{\mathrm{o}}$-Segal

iso-morphism. Then

a

pointwisely defined annihilation operator, which is also called Hida’s

differential operator and is denoted by $\partial_{t}$, becomes

a

continuous operator

on

$(E)$; and

a

pointwisely defined creation operator $\partial_{t}^{*}$ is

a

continuous operator

on

$(E)^{*}$

.

In

a

series

of

works $[10]-[12]$

we

have established

a

systematic theory

of

operators

on

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

$\int_{T^{\iota+m}}\kappa(_{S\cdots,S}1,l, t1, \cdots,t_{m})\partial*\ldots\partial_{S_{l}}^{*}\partial_{t_{1}}\cdots\partial t_{m}ds_{1}\cdots dS\iota dt_{1}\cdots dt_{m}S1$

’ (1)

where $\kappa$

is

a

distribution in $l+m$ variables. It should be emphasized strongly that

an

integral kernel $\kappa$

can

be

a

distribution. In fact, the composition $\partial_{s_{1}s}^{*}\ldots\partial^{*}\mathrm{t}\partial t_{1}\ldots\partial_{t_{m}}$ is well

defined (namely, normally ordered product) and becomes

a

continuous operator from$(E)$

into $(E)^{*}$

.

Moreover, the dependenceof the parameters $s_{j}$ and $t_{k}$ is smooth enough.

The kernel distribution $\kappa$ in (1) being regarded

as a

scalar operator-valued distributon,

we

are

led quite naturally to

a

generalization with

an

integral kernel being

an

operator-valued distribution. In this note

we

shall introduce

an

operator in the following form:

Of course, this is a formal (but sometimes very descriptive) expression. For the precise

definition

we

need the characterization theoremfor operatorsymbols and

some

properties

ofoperator-valued distributions. Those results

are

obtained in $[10]_{-}[12]$.

As application

we

discuss

an

operator-valued (or quantum) stochasticprocess of

Hitsuda-Skorokhod 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]. Furtherdetailed study in this direction will appearin

a

forthcoming

paper.

ACKNOWLEDGEMENTS. I

am very

grateful to Professors I. Kubo and H. Watanabe

for

interesting 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 $T$ be a topological space with

a

Borel

measure

_$\nu(dt)=dt$

which is thought of

as a

timeparameter space when it is

an

interval,

or

more

generally

as

a

field parameter space. Given

a

positive selfadjoint operator $A$

on

the real Hilbert space

$H=L^{2}(T, \nu;\mathbb{R})$ with Hilbert-Schmidt inverse,

one

may form

a

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

in the standard manner; namely, $E$ is the $C^{\infty}$-domain of $A$ equipped with the Hilbertian

norms

$|\xi|_{\mathrm{p}}=|A^{p}\xi|_{0}$ , $\xi\in H$

,

$p\in \mathbb{R}$

,

where $|\cdot|_{0}$ is the

norm

of $H$

.

Such

a

countably Hilbert space is called

a

standard

CH-space,

see

[11]. Since $A^{-1}$ is of Hilbert-Schmidt type, _$E$ becomes

a

nuclear space. The

canonical bilinear form

on

$E^{*}\cross E$ and the real inner product of $H$

are

denoted by the

same

symbol $\langle\cdot, \cdot\rangle$ without contradiction.

One

can

thinkof$E$and $E^{*}$

as

spaces oftest and generalizedfunctions

on

$T$, respectively.

In order to keep the delta functions $\delta_{t}$ within

our

discussion

we

assume:

(H1) for each $\xi\in E$thereexists

a

uniquecontinuous function$\xi \mathrm{o}\mathrm{n}T\sim$such that _{$\xi(t)=\xi(t)\sim$}

for

v-a.e.

$t\in T$;

(H2) for each $t\in T$

a

linearfunctional $\delta_{t}$ : _{$\xirightarrow\xi(t),$}$\xi\sim\in E$, is continuous, i.e., _{$\delta_{t}\in E^{*};$}

(H3) the map $trightarrow\delta_{t}\in E^{*},$ $t\in T$, is continuous with respect to the strong dual topology

$\mathrm{o}\mathrm{f}E^{*}$

.

From

now

on

we

always

assume

that every element in $E$ is

a

continuous function

on

$T$

and do not

use

the symbol $\xi\sim$. For another

reason we

need

one

more

assumption:

(S) inf$\mathrm{S}\mathrm{p}\mathrm{e}\mathrm{C}(A)>1$

.

We then put

$\delta=||A^{-1}||_{\mathrm{H}\mathrm{S}}<\infty$, $\rho=||A^{-1}||_{\mathrm{o}\mathrm{P}}=(\inf \mathrm{S}\mathrm{p}\mathrm{e}\mathrm{c}(A))^{-1}$ .

The obvious inequalities

$0<\rho<1$; $|\xi|_{\mathrm{P}}\leq\rho^{q}|\xi|_{p+}q$

are

used throughout with

no

special notice.

The Gaussian measure $\mu$ is by definition

a

probability

measure on

$E^{*}$ of which

charac-teristic function is:

$\exp(-\frac{1}{2}|\xi|_{0}2)=\int_{E^{*}}e^{i}(x,\epsilon\rangle\mu(dX),$ _{$\xi\in E$}.

The probability space $(E^{*}, \mu)$ is called

a

Gaussian space. We put

$(L^{2})=L^{2}(E^{*}, \mu;\mathbb{C})$

for simplicity.

The canonical bilinear form

on

$(E^{\otimes n})^{*}\cross(E^{\otimes n})$ is denoted by $\langle\cdot, \cdot\rangle$ again and its $\mathbb{C}-$

bilinear extension to $(E_{\mathbb{C}}^{\otimes n})^{*}\cross(E_{\mathbb{C}}^{\otimes n})$ is also denoted by the

same

symbol. For $x\in E^{*}$ let

:$x^{\otimes n}$: be defined

as a

unique element in

$(E^{\otimes n})_{\mathrm{S}}*\mathrm{y}\mathrm{m}$satisfying

$\phi_{\xi}(X)\equiv\sum_{=n0}^{\infty}\langle:x^{\otimes n}:,$ $\frac{\xi^{\otimes n}}{n!}\rangle=\exp(\langle x, \xi\rangle-\frac{1}{2}\langle\xi, \xi\rangle)$, $\xi\in E_{\mathbb{C}}$

.

(3)

This “normalized” exponential function $\phi_{\xi}$ is called

an

exponential vector. In particular,

$\phi 0$ is the

vacuum.

As is well known, each $\emptyset\in(L^{2})$ is expressed in the

following

form: $\phi(x)=\sum_{=n0}^{\infty}\langle:X:,$$fn\otimes n\rangle$, $x\in E^{*}$, $f_{n}\in H_{\mathbb{C}}^{\otimes n}\wedge$, (4)

where each $xrightarrow$ $\langle:x^{\otimes n}:, f_{n}\rangle$ and the

convergence

of the series

are

understood in the

$L^{2}$

-sense.

Expression (4) is referred to

as

the Wiener-It\^o expansion of $\phi$. In that case,

$|| \phi||_{0}^{2}\equiv\int_{E^{\mathrm{e}}}|\phi(x)|^{2}\mu(dX)=n=0\sum n!|f_{n}|^{2}\infty 0^{\cdot}$ (5)

Thus

we

have

a

unitary isomorphism between $(L^{2})$ and the Boson Fock space

over

$H_{\mathbb{C}}$,

which is the celebrated $\mathrm{W}\mathrm{i}\mathrm{e}\mathrm{n}\mathrm{e}\mathrm{r}-\mathrm{I}\mathrm{t}_{\hat{\mathrm{O}}}$-Segal isomorphism.

The second quantized operator of $\mathrm{A}$, denoted by

$\Gamma(A)$, is

an

operator in $(L^{2})$

defined

by

$\Gamma(A)\phi(x)=n\sum^{\infty}\langle:X^{\otimes}:,$_{$A^{\otimes}nfn\rangle=0n$} ,

where $\phi\in(L^{2})$ is given

as

in (4). Equipped with the maximal domain, $\Gamma(A)$ becomes

a

positive selfadjoint operator

on

$(L^{2})$ and

we

obtain

a

standard $\mathrm{C}\mathrm{H}$-space which will be

denoted by $(E)$. That $\Gamma(A)$ admits

a

Hilbert-Schmidtinverse is guaranteed byhypothesis

(S). Therefore, $(E)$ becomes

a

nuclear Fr\’echet space and

we come

to

a

complex Gelfand

triple:

$(E)\subset(L^{2})=L^{2}(E^{*}, \mu;\mathrm{c})\subset(E)^{*}$.

Elements in $(E)$ and $(E)^{*}$

are

called

a

test (white noise)

functional

and

a

generalized

(white noise) functional, respectively. We denote by $\langle\langle\cdot, \cdot\rangle\rangle$ the canonical bilinear form

on

$(E)^{*}\cross(E)$ and by $||\cdot||_{p}$ the

norm

introduced from $\Gamma(A)$, namely,

where$\phi$and $(f_{n})_{n=0}^{\infty}$

are

related

as

in (4). Thus (5) is

a

special

case

of(6). As is easily

seen

from (6), $\emptyset\in(L^{2})$ belongs to _$(E)$ if and only if$f_{n}\in E_{\mathbb{C}}^{\otimes n}\wedge$ for all

$n$ and $\sum_{n=0}^{\infty}n!|f_{n}|_{p}^{2}<\infty$

for all $p\geq 0$.

We

use a

similar (but formal) expressionfor

a

generalized whitenoise functional. Every

$\Phi\in(E)^{*}$ is written

as

$\Phi(x)=\sum^{\infty}\langle n=0:X^{\otimes}:n,$ $F_{n\rangle}$ , (7)

where $F_{n}\in(E_{\mathbb{C}}^{\otimes n})^{*}\mathrm{s}\mathrm{y}\mathrm{m}$ and

$|| \Phi||_{-p}^{2}=n\sum_{=0}^{\infty}n!|F_{n}|_{-p}^{2}$ . (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 expansionof $\Phi$. In that case,

$\langle\langle\Phi, \phi\rangle\rangle=\sum_{n=0}^{\infty}n!\langle F_{n},$ $fn)$ ,

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

as

in (4).

2

Integral 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 delta

functions $\delta_{t}$

are

elements in $E^{*}$ by hypotheses $(\mathrm{H}1)-(\mathrm{H}3)$,

we

may define

$\partial_{t}=D_{\mathit{5}_{t}}$, $t\in T$

.

This is called Hida’s

_differential

operator. Obviously, $\partial_{t}$ is

a

rigorously

defined

annihi-lation operator at

a

point $t\in T$. It should be

therefore

emphasized that $\partial_{t}$ is not

an

operator-valued distribution but

a

continuous

operator

for itself.

The creation operator

is by definition the adjoint $\partial_{t}^{*}\in \mathcal{L}((E)^{*}, (E)^{*})$ and

we come

to the so-called canonical

commutation relation:

$[\partial_{S}, \partial_{t}]=0$, $[\partial_{s}^{*}, \partial_{t}^{*}]=0$, $[\partial_{s}, \partial_{t}^{*}]=\delta_{s}(t)I$, $s,t\in T$. (10)

The last relation is understood in

a

generalized

sense.

For $\phi,$$\psi\in(E)$ let

$\eta_{\phi,\psi}$ be

a

function

on

$T^{l+m}$ defined by

$\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$

.

(11)

Then $\eta_{\phi,\psi}\in E_{\mathbb{C}}^{\otimes(lm}+$) and $(\phi, \psi)rightarrow\langle\kappa, \eta_{\phi},\psi\rangle$ is

a

continuous bilinear form

on

$(E)$

for

$\mathrm{a}\mathrm{n}\mathrm{y}--\kappa\in(E_{\mathbb{C}}^{\otimes(l+m)})^{*}$

.

By general theory there exists

a

unique continuous linear operator

$rightarrow\iota_{m},(\kappa)\in \mathcal{L}((E), (E)^{*})$ such that

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

This suggests

us

to employ

a

formal integral expression:

–lm–,$( \kappa)=\int_{T^{l+m}}\kappa(S1, \cdots, S_{l,1}t, \cdots, t_{m})\partial_{s1s}^{*}\ldots\partial^{*}\partial_{t_{1}}\cdots\partial tmdS_{1}\cdots dS_{l}dt\iota 1\ldots dt_{m}$

.

We call $–l.m-(\kappa)$

an

integral kernel operator with kernel distribution $\kappa$

.

It is noteworthy

that $–l,m-(\kappa)$ is defined for any $\kappa\in(E_{\mathbb{C}}^{\otimes(+)}\iota m)^{*}$ and becomes

a

continuous operator in

$L((E), (E)^{*})$. For any $p>0$ with $|\kappa|_{-p}<\infty$

we

have

$||_{-}^{-}-\prime_{m},(\kappa)\phi||-p\leq Cl,m;p|\kappa|_{-}p||\emptyset||_{p}$ , $\phi\in(E)$, (13)

where

$c_{l,m;p}=\rho-\mathrm{p}$

(llm)

1m/2

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

)$/2$

This estimate is useful. Recall that $|\kappa|_{-p}<\infty$ for all sufficiently large $p>0$

.

The kernel distribution is not uniquely determined due to relation (10); however, for

the uniqueness

we

only need to restrict ourselves to the subspace $(E_{\mathbb{C}}^{\otimes(l+}m))_{\mathrm{s}}^{*}\mathrm{y}\mathrm{m}(l,m)$

of

all

$\kappa\in(E_{\mathbb{C}}^{\otimes(+m)})^{*}l$ which is symmetric with respect to the first $l$ and the last _$m$ variables

independently.

3

Symbol and Fock

expansion

$\mathrm{F}\mathrm{o}\mathrm{r}---\in \mathcal{L}((E), (E)^{*})$

a

function

on

$E_{\mathbb{C}}\cross E_{\mathbb{C}}$ defined by

$—\wedge(\xi, \eta)=\langle\langle_{-}^{-}-\phi_{\xi}, \phi_{\eta}\rangle\rangle$

,

$\xi,$$\eta\in E_{\mathbb{C}}$, (14)

is called the symbol $\mathrm{o}\mathrm{f}---$

.

Since the exponential vectors $\{\phi_{\xi;}\xi\in E_{\mathbb{C}}\}$

spans

a

dense

sub-space of $(E)$, the synlbol

recovers

the operator uniquely. For

an

integral kernel operator,

$–l,m-\overline{(}\kappa)(\xi, \eta)=\langle\kappa,$ $\eta^{\otimes l}\otimes\xi^{\otimes}m\rangle e^{\mathrm{t}\xi,\eta\}}$, (15)

or

equivalently,

$\langle\langle_{-l,m}^{-}-(\kappa)\emptyset\epsilon, \phi_{\eta}\rangle\rangle=\langle\langle\langle\kappa,$ $\eta\otimes\otimes l\xi\otimes m\rangle\phi_{\xi},$ $\phi_{\eta}\rangle\rangle$ , (16)

where $\xi,$$\eta\in E_{\mathbb{C}}$ and $\kappa\in E_{\mathbb{C}}^{\otimes 1^{\iota}+}m$) It is

straightforward

to

se.e

that $\ominus=---\wedge,$ $—\in$

$\mathcal{L}((E), (E)^{*})$, possesses the following two properties:

(O1) For any $\xi,$ $\xi_{1},$

$\eta,$$\eta_{1}\in E_{\mathrm{C}}$

,

the function

$z,$ $wrightarrow\ominus(z\xi+\xi 1, w\eta+\eta_{1})$, $z,$$w\in \mathbb{C}$,

is entire holomorphic;

(02) There exist constant numbers $C\geq 0,$ $K\geq 0$ and $p\in \mathbb{R}$such that

More important is that the

converse

is also true. Theorem

$bl3.1$ Any

$\mathbb{C}$-valued

function

$\Theta$ on

$E_{\mathbb{C}}\cross E\mathrm{c}_{\underline{\wedge}}satisfyin=\ominus g$conditions

$(Ol)$ and $(O\mathit{2})$

is the symbol

_of

an

$operat_{\mathit{0}}r---\in \mathcal{L}((E), (E)^{*}),$ $i.e.$, –

$\kappa_{l,m}\in(E_{\mathbb{C}}^{\otimes}m)_{\mathrm{S}}*\mathrm{C}(\iota,m)\mathrm{S}\mathrm{u}\mathrm{h}\mathrm{t}\mathrm{I}\mathrm{n}\mathrm{f}\mathrm{a}\mathrm{C}\mathrm{t},\ovalbox{\tt\small REJECT}_{+}^{\mathrm{i}\mathrm{v}}\mathrm{e})\mathrm{n}\mathrm{S}\mathrm{u}\mathrm{c}\mathrm{y}\mathrm{m}\mathrm{h}\mathrm{a}\mathrm{f}\mathrm{u}\mathrm{n}\mathrm{C}\mathrm{t}\mathrm{i}\mathrm{h}\mathrm{a}\mathrm{t}\mathrm{o}\mathrm{n}\ominus$

, there exists

a

unique family of kernel

distributions

$\ominus.(\xi, \eta)=,\sum_{0\iota_{m=}}^{\infty}\langle\langle---l,m(\kappa_{l,m})\phi\epsilon, \phi_{\eta}\rangle\rangle$ , $\xi,$$\eta\in E_{\mathbb{C}}$.

Moreover, the series

$– \phi-=l,\sum_{m=0}^{\infty}---_{\iota_{m}},(\kappa_{l,m})\emptyset$, $\phi\in(E)$, (17)

converges

in $(E)^{*}$, and thereby

we

obtain _{$—\in \mathcal{L}((E), (E)^{*})$} of which symbol is $O-$

.

In

particular, the symbol $\mathrm{o}\mathrm{f}---\in \mathcal{L}((E), (E)^{*})$ satisfying (O1) and (02), the above argument

reproduces

an

$\mathrm{o}\mathrm{p}\mathrm{e}\mathrm{r}\mathrm{a}\mathrm{t}\mathrm{o}\mathrm{r}---\mathrm{i}\mathrm{n}$terms of integral kernel operators. Expression (17) is called

the Fock expansion$\mathrm{o}\mathrm{f}---$

.

In

some

practical problems operators

on

Fock space

are

only

defined

on

the exponential

vectors $\{\emptyset\epsilon;\xi\in E_{\mathbb{C}}\}$ due to the fact that they

are

linearly independent. Theorem 3.1 is

therefore 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

of $\mathcal{L}((E), (E))$ which

is also important from

some

applications. For complete information

see

[11].

4

Operator-valued

_{distributions}

In [12]

we

studied $\mathcal{L}(\mathcal{E}, \mathcal{E}^{*})$-valued distributions in general, where $\mathcal{E}$ is

a

standard

CH-space. Here

we

recapitulate

some

results for $\mathcal{E}=(E)$

.

Let $\{e_{j}\}_{j=}^{\infty}0$ be the normalized eigenfunctions

of

the operator $A$. For $\mathrm{i}=(\dot{i}_{1}, \cdots, i_{l})$ and

$\mathrm{j}=(j_{1}, \cdots,j_{m})$

we

put

$e(\mathrm{i})=e_{i_{1}}\otimes\cdots\otimes e_{i}l$

’ $e(\mathrm{j})=e_{j_{1}}\otimes\cdots\otimes e_{j_{m}}$

.

For

a

linear map $L:E_{\mathbb{C}}^{\otimes}(\iota+m)arrow \mathcal{L}((E), (E)^{*})$ and

$p,$$q,$ $r,$$s\in \mathbb{R}$

we

put

$||L|| \iota_{mp,q;},;r,s=\sup\dagger\sum_{\mathrm{i}_{\dot{\mathrm{d}}}}|\langle\langle L(e(\mathrm{i})\otimes e(\mathrm{j}))\emptyset, \psi)\rangle|2|e(\mathrm{i})|_{p}^{2}|e(\mathrm{j})|^{2}q$ ; $\phi,\psi\in(||\emptyset|||\psi|^{1}|-r\leq 1-s\leq 1E)\}^{1/2}$

By

definition

for

any

$p,$ $q,$ $r,$$s\in \mathbb{R}$

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

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

For brevity

we

put

$||L||_{p}=||L||_{l,p}m;p,p;p,$ _’ $||L||_{l,;}mp,q=||L||_{l,;p,q}m;p,q$

.

The next result will be useful, for the proof

see

[12].

Proposition 4.1 Fora linear map $L:E_{\mathbb{C}}^{\otimes \mathrm{t}^{l}+m)}arrow \mathcal{L}((E), (E)^{*})$ the$foll_{ow}ingfour$

condi-tio$ns$ are equivalent:

(i) $L\in \mathcal{L}(E_{\mathbb{C}}^{\otimes}m)\mathcal{L}\{l+,((E), (E)*))$;

(ii) $\sup\{|\langle\langle L(\eta)\phi, \psi\rangle\rangle|;\eta\in E_{\mathbb{C}}\phi,\psi\in\otimes(E^{+})(lm,),$ $||\phi|||\eta|_{p}\leq \mathrm{P}\leq 1,$$|1|\psi||_{p}\leq 1\}<\infty$

for

some

$p\geq 0_{j}$

(iii) $||L||_{-p}<\infty$

for

some

$p\geq 0_{i}$

(iv) $||L||_{l,m;p},q;r,S<\infty$

for

some

$p,$ $q,$ $r,$$s\in \mathbb{R}$.

In that case,

_for

any$p,$ $q,$ $r,$$s\in \mathbb{R}$ we have

$|\langle\langle L(\eta)\emptyset, \psi\rangle\rangle|\leq||L||_{l,r}m;-p,-q;-,-S|\eta|l,m;p,q||\phi||S||\psi||_{r}$ , (20)

and

$||L(\eta)\emptyset||-r\leq||L||l,m;-p,-q;-r,-s|\eta|_{l,q}m;p,||\phi||S$ , (21)

where $\eta\in E_{\mathbb{C}}^{\otimes(+\rangle}\iota m,$ $\phi,$$\psi\in(E)$.

Each $L\in \mathcal{L}(E_{\mathbb{C}}^{\otimes(m)}+, \mathcal{L}(l(E), (E)^{*}))$ is justifiably called

an

$\mathcal{L}((E), (E)^{*})$-valued

distri-bution

on

$T^{l+m}$. If $\mathcal{L}((E), (E)^{*})$

were

a

Fr\’echet space,

one

would have

a

canonical

iso-morphism

$\mathcal{L}(E_{\mathbb{C}}^{\otimes(m)}, \mathcal{L}+((\iota E), (E)^{*}))\cong(E_{\mathbb{C}}^{\otimes(+}\iota m))*\otimes \mathcal{L}((E), (E)^{*})$

by the kernel theorem; however, $\mathcal{L}((E), (E)^{*})$ is not

a

Fr\’echet space. (It is known that

$\mathcal{L}(\mathcal{E}, \mathcal{E}^{*})$ is Fr\’echet if and only if

$\mathcal{E}$

is

a

Hilbert space.)

5

Generalization of integral kernel operators

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

we

associate

an

operator _{$—\in \mathcal{L}((E), (E)^{*})$} by

the formula:

$\langle(_{-}^{-}-\phi_{\xi}, \phi_{\eta}\rangle\rangle=\langle\langle L(\eta^{\otimes l}\otimes\xi\otimes m)\phi\epsilon,$ $\phi_{\eta}\rangle\rangle,$ $\xi,$$\eta\in E_{\mathbb{C}}$. (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

$|\ominus(\xi, \eta)|$ $\leq$ $||L||_{-\mathrm{p}}|\eta^{\otimes l}\otimes\xi^{\otimes}m|_{p}||\phi_{\xi}||_{p}||\phi\eta||_{p}$

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

$\ominus$ is the symbol of

an

$\mathrm{o}\mathrm{p}\mathrm{e}\mathrm{r}\mathrm{a}\mathrm{t}\mathrm{o}\mathrm{r}---\in \mathcal{L}((E), (E)^{*})$; namely, there exists

a

unique operator $—\in \mathcal{L}((E), (E)^{*})$ satisfying (22). It is reasonable to write

$—= \int_{T}\partial_{s}^{*}1\ldots\partial_{s\iota}*L(s1, \cdots, s\iota, t_{1}, \cdots, tm)\partial_{t_{1}}\cdots\partial tmd_{S_{1}}\cdots dS_{l}dt1\ldots dt_{m}$

.

(24)

The above constructed $\mathrm{o}\mathrm{p}\mathrm{e}\mathrm{r}\mathrm{a}\mathrm{t}\mathrm{o}\mathrm{r}---\mathrm{i}\mathrm{s}$

a

generalizationof

an

integral kernel operator

intro-duced in

\S 2,

compare (22) and (16).

This genera,lization

occurs

in

an

integral kernel operator. Let $\kappa\in(E_{\mathbb{C}}m))_{\mathrm{s}\mathrm{y}}^{*}\otimes(l+\mathrm{m}(l,m)$ and

consider

an

integral kernel$\mathrm{o}\mathrm{p}\mathrm{e}\mathrm{r}\mathrm{a}\mathrm{t}_{\mathrm{o}\mathrm{r}}--_{l,m}-(\kappa)$ . To

go

further

we

need contraction of tensor

products. For $g_{l}\in E_{\mathbb{C}}^{\otimes l}$ and $g_{n}\in E_{\mathbb{C}}^{\otimes n}$

we

define $\kappa\otimes^{l}(g_{l}\otimes g_{n})\in(E_{\mathbb{C}}^{\otimes(m+n)})*\mathrm{a}\mathrm{s}$

a

unique

element satisfying

$\langle\kappa\otimes^{l}(gl\otimes gn), \zeta\rangle=\langle\kappa\otimes g_{n}, g_{l}\otimes\zeta\rangle$

,

$\zeta\in E_{\mathbb{C}}^{\otimes()}m+n$.

Then $\kappa\otimes^{l}g$ is defined for any $g\in E_{\mathbb{C}}^{\otimes(l)}+m$ by continuity and is called

a

left contraction.

Moreover, it is easily verified that

$|F\otimes^{\iota_{g}}|_{-p}\leq\rho^{2pn}|F|-p|g|p$ ’

$F\in(E_{\mathbb{C}}^{\otimes(+})\iota m)*$, $g\in E_{\mathbb{C}}^{\otimes(+n)}l$

.

(25)

The right contraction $\kappa\otimes_{l}g$ is similar. For detailed argument

see

[11].

Lemma 5.1 Fix integers $0\leq\alpha\leq l$ 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

$|\langle\langle L_{0}(\eta_{1}, \cdots, \eta\alpha’\xi_{1}, \cdots, \xi\beta)\emptyset, \psi\rangle\rangle|\leq c_{l_{-}\alpha},m-\beta,p|\lambda|_{-}p||\emptyset||_{p}||\psi||p$ . (27)

Given bounded subsets $B_{1},$$B_{2}\subset(E)$,

we see

from (26) and (27) that

$\sup$ $|\langle\langle L_{0}(\eta_{1}, \cdots,\eta\alpha’\xi_{1}, \cdots, \xi\beta)\emptyset, \psi\rangle\rangle|$

$\emptyset\in B_{1,\psi\in B_{2}}$

which implies the desired continuity of$L_{0}$

.

By Lemma 5.1 there exists$L\in \mathcal{L}(E_{\mathbb{C}}^{\otimes\{\beta)}\alpha+, \mathcal{L}((E), (E)^{*}))$ such that

qed

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

.

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 $0\leq\beta\leq m$. Given $\kappa\in$

$(E_{\mathbb{C}}^{\otimes \mathrm{t}l+m)})^{*}$ let $L\in \mathcal{L}(E_{\mathbb{C}}^{\otimes \mathrm{t}}\alpha+\beta),$_{$\mathcal{L}((E), (E)^{*}))$} be

defined

as

in (28). Then,

$–l,m-( \kappa)=\int_{T^{\alpha+\beta}}\partial*\ldots\partial_{S_{q}}^{*}L(_{S_{1,\alpha}}s_{1}\ldots,)s,t_{1}, \cdot*\cdot, t_{\beta}\partial_{t_{1}t}\ldots\partial ds_{1}\cdots dS\beta\alpha dt_{1}\cdots dt_{\beta}$

.

PROOF. The symbol oftheright hand side is $\langle\langle L(\eta^{\otimes\alpha}\otimes\xi^{\otimes\beta})\phi\xi, \phi_{\eta}\rangle\rangle$by definition (22).

In view of (28)

we

obtain

$\langle\langle L(\eta^{\otimes\otimes}\otimes\alpha\xi\beta)\phi\epsilon,$ $\phi_{\eta}\rangle\rangle$ _$=$ $\langle\langle_{-l-}^{-}-\alpha,m-\rho((\kappa\otimes\beta\xi^{\otimes\beta})\otimes^{\alpha\otimes}\eta)\alpha\phi\xi,$$\phi\eta\rangle\rangle$

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

$=$ $\langle\kappa,$ $\eta\xi\otimes l_{\otimes}\otimes m\rangle e^{(}\xi,\eta\rangle$.

The last expression coincides with the symbol $\mathrm{o}\mathrm{f}_{-l,m}--(\kappa)$ by (15) and hence follows the

assertion. qed

The above result is essential to discuss “canonical form” of

an

adapted operator-valued

process. This topic will be discussed in

a

forthcoming paper.

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 $L$ should possess

a

stronger property that $L$ is continuously extended

to

a

linear map from $E_{\mathbb{C}}^{*}$ into $\mathcal{L}((E), (E)^{*})$. Note the natural inclusion relation

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

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

we

write $L_{s}=L(\delta_{s})$ for simplicity. Then $\{L_{s}\}$ is regarded

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

for

any $f\in E_{\mathbb{C}}^{*}$ there exists an operator

$M_{f}\in \mathcal{L}((E), (E)^{*})$ such that

$\langle\langle M_{f}\phi C, \phi_{\eta}\rangle\rangle=\langle\langle L(f\eta)\emptyset\epsilon, \phi_{\eta}\rangle\rangle$ , $\xi,$$\eta\in E_{\mathbb{C}}$

.

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

.

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

$(\langle L(f)\emptyset, \psi\rangle\rangle=(f, L^{*}(\phi\otimes\psi)\rangle,$ $f\in E_{\mathbb{C}}^{*}$, $\phi,$$\psi\in(E)$

.

By continuity, for any $p\geq 0$ thereexist $q\geq 0$ and $B_{p,q}\geq 0$ such that

$|L^{*}(\emptyset\otimes\psi)|p\leq B_{p},q||\phi||_{\rho+}q||\psi||_{p+q}$, $\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)

we

may 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

$|\ominus_{f}(\xi, \eta)|\leq C|f|_{-p}\exp Ic(|\xi|_{p+S^{+}}^{2}|\eta|_{p+S}2)$ , $f\in E_{\mathbb{C}}^{*}$, $\xi,$$\eta\in E_{\mathbb{C}}$. (31)

Hence by the characterization theorem (Theorem 3.1), for any $f\in E_{\mathbb{C}}^{*}$ there exists

an

operator $\Lambda/I_{j}\in \mathcal{L}((E), (E)^{*})$ such that

$\langle\langle M_{f}\phi_{\xi}, \phi_{\eta}\rangle\rangle=\ominus_{f}(\xi, \eta)=\langle\langle L(f\eta)\phi\epsilon, \phi_{\eta}\rangle\rangle$

,

$\xi,$$\eta\in E_{\mathbb{C}}$.

Obviously, $f\vdasharrow \mathbb{J}l_{f}$ is linear. Inequality (31) implies the continuity

on

the Hilbert space

$\{f\in E_{\mathbb{C}}^{*}; |f|_{-p}<\infty\}$. Since $E_{\mathbb{C}}^{*}$ is the inductivelimit of such Hilbert spaces,

we

conclude

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

$\Omega_{t}\equiv\int_{0}^{t}\partial_{s}^{*}L_{s}d_{S}$, _{$t\geq 0$},

which forms

a

one-parameter family of operators in $\mathcal{L}((E), (E)^{*})$

.

This is called

an

operator-valued integral

_of

Hitsuda-Skorokhod type. To be

sure

we

rephrase the

defini-tion:

$\langle\langle\Omega_{t}\phi\epsilon, \phi_{\eta}\rangle\rangle=\langle\langle L(1[0,t]\eta)\phi\epsilon, \phi_{\eta})\rangle$, $\xi,$$\eta\in E_{\mathbb{C}}$

.

(32)

It is interesting to observe how

our

operator-valued process $\{\Omega_{t}\}$ generalizes $\mathrm{t}_{\mathrm{J}}\mathrm{h}\mathrm{e}$

Hitsuda-Skorokhod integral.

For that purpose

we

quote the definition of the Hitsuda-Skorokhod integral following

[4]. Let $\Phi_{t}\in(E)^{*},$ $t\geq 0$, be given. Since $\partial_{t}^{*}\in \mathcal{L}((E)^{*}, (E)^{*})$ for any $t$, for

any

_$\phi\in(E)$

one

obtains

a

function: $trightarrow\langle\langle\partial_{t}^{*}\Phi_{t}, \phi\rangle\rangle$. Assume that the function is measurable and

$\int_{0}^{t}|\langle\langle\partial^{*}\Phi ss’\phi\rangle\rangle|ds<\infty$, $t\geq 0$

.

Then, it is proved that there exists $\Psi_{t}\in(E)^{*},$ $t\geq 0$, uniquely such that

$\langle\langle\Psi_{t}, \phi\rangle\rangle=\int_{0}^{t}\langle\langle\partial_{s}*\Phi_{s}, \phi\rangle\rangle ds$ , $\phi\in(E)$.

The above obtained $\Psi_{t}$ 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

integral coincides with the usual It\^o integral when the integrand $\{\Phi_{t}\}$ is

an

adapted $L^{2_{-}}$

function with respect to the filtration generated by the Brownian motion

$B_{t}(x)=\langle x,$ $1_{[0,t]}\rangle$, $x\in E^{*}$, $t\geq 0$

.

In this connection

see

also [7], [8].

We need

one more

remark. Each $\Phi\in(E)^{*}$ gives rise to

a

continuous operator in

$\mathcal{L}((E), (E)^{*})$ by lnultiplicationsince $(\phi, \psi)rightarrow\phi\psi$is

a

continuous bilinear map from $(E)\cross$

$(E)$ into $(E)$. This identification extends to

a

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

Now suppose

we

are

given $\Phi\in \mathcal{L}(E_{\mathbb{C}}^{*}, (E)^{*})$. Let

$\tilde{\Phi}$

denote the corresponding

multipli-cation operator, i.e., $\tilde{\Phi}\in \mathcal{L}(E_{\mathbb{C}}^{*}, \mathcal{L}((E), (E)^{*}))$. Then

one

has

an

operator-valued integral

of Hitsuda-Skorokhod type:

as

well

as

the Hitsuda-Skorokhod integral in the original

sense:

$\Psi_{t}=\int_{0}^{t}\partial_{s}*\Phi_{s}d_{S}$, _{$t\geq 0$}. (34)

In fact, since the both maps $trightarrow\delta_{t}\in E^{*}$ and $trightarrow\partial_{t}\phi\in(E)$

are

continuous,

so

is

$trightarrow\langle\langle\partial_{t}^{*}\Phi_{t}, \emptyset\rangle\rangle$

.

Therefore $\Psi_{t}$ is well defined.

Theorem 6.2 Forany$\Phi\in \mathcal{L}(E_{\mathbb{C}}^{*}, (E)^{*})$ let $\Omega_{t}$ be the operator-valued integral

of

Hitsuda-Skorokhod type

_defined

as

in (33) and let $\Psi$ be the Hitsuda-Skorokhod integral in the

original sense

_defined

as in (34). Then,

$\Psi_{t}=\Omega_{t}\phi 0$, $t\geq 0$,

where $\phi_{0}$ is the

vacuum.

PROOF. By definition (32)

we

have

$\langle\langle\Omega_{t}\emptyset 0, \phi\eta\rangle\rangle=\langle\langle\tilde{\Phi}(1_{[0,t}]\eta)\phi 0,$ $\emptyset\eta\rangle\rangle=\langle\langle\Phi(1_{[}0,t]\eta),$ $\phi_{\eta\rangle\rangle}$ .

In terms of the adjoint operator $\Phi^{*}\in \mathcal{L}((E), E_{\mathbb{C}})$ the last expression becomes

$\langle\langle\Phi(1[0,t]\eta),$ $\emptyset\eta\rangle\rangle=\langle 1_{[t]\eta}0,,$ $\Phi^{*}\phi\eta\rangle=\int_{0}^{t}\eta(s)(\Phi*\phi_{\eta})(S)d_{S}$

.

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

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