数理解析研究所講究録
923 巻 1995 年 33-51

33

ON-MEHLER TRANSFORMS
AND INFINITESIMAL GENERATORSIN WHITE NOISE CALCULUS*)$\mathrm{P}\mathrm{S}\mathrm{E}\mathrm{U}\mathrm{D}\mathrm{o}_{-}\mathrm{p}\mathrm{o}\mathrm{U}\mathrm{R}\mathrm{I}\mathrm{E}\mathrm{R}$ISAMU$\mathrm{e}$$\mathrm{D}\hat{\mathrm{O}}$KU (道工勇)Department of Mathematics
Saitama University
Urawa 338 Japan-mail: doku@sacs.sv.saitama-u.ac.jp\S 1. Introduction
The study of the Fourier transformin white noise calculus was initiated andhas been developed to a mature level by H.-H. Kuo $[16,17]$ (also. While, theis a kind of generalization of
Fourier-Mehler transform(also [11]), which
furnishes the theory of infinite dimensional Fourier transforms in white noise spacewith adequately fruitful and profitable ingradients.
In this article we introduce-Mehler (PFM for short) transformhaving quite similar nice properties as the Fourier-Mehler transform possesses. It
was originally defined in [5] and used for application to abstract equations in infinite
dimensional spaces. In connection with other Fourier type transforms in white noise
analysis, we can compute the infinitesimal generator of the PFM transform directly
and show that our-Mehler transform enjoys intertwining properties.
We shall state the characterization theorem for PFM transforms, which is one ofour main results in this article. The Fock expansion of PFM transform can bederived as well. Lastly we shall introduce a generalization idea of PFM transformand investigate some properties that the generalized transform should satisfy.
The-Mehler transform is a very important and interesting operator in the standpoint of how to express the solutions for the Fourier-transformed
abstract Cauchy problems ([5,6]; see also [4,8]).
In [1] they have studied the two dimensional complex Lie group explicitly andsucceeded in describing every one parameter subgroup with infinitesimal generator$( \frac{2a+b}{2})\Delta_{G}+bN$ , where $N$ is the number operator andis the Gross Laplacian.
Furthermore, one can find in [24] another related work, especially on a systematic
study of Lie algebras containing infinite dimensional Laplacians.
We are able to state our results in the general setting (e.g., [23]; see also [7])of white noise analysis. As a matter of fact, almost all statements in our theory$\mathcal{F}$$[19]\rangle$$\mathcal{F}[18]$$\mathcal{F}_{\theta}$$\mathrm{P}\mathrm{s}\mathrm{e}\mathrm{u}\mathrm{d}_{0}-\mathrm{F}\mathrm{o}\mathrm{u}\mathrm{r}\mathrm{i}\mathrm{e}\mathrm{r}$$\mathrm{p}\mathrm{s}\mathrm{e}\mathrm{u}\mathrm{d}\mathrm{o}- \mathrm{F}_{\mathrm{o}\mathrm{u}\mathrm{r}\mathrm{i}}\mathrm{e}\mathrm{r}$$\mathrm{p}_{\mathrm{S}}\mathrm{e}\mathrm{u}\mathrm{d}\mathrm{o}-\mathrm{F}_{\mathrm{o}\mathrm{u}\mathrm{r}\mathrm{i}}\mathrm{e}\mathrm{r}$$\mathcal{G}$$\Delta_{G}$Research is supported in part by JMESC Grant-in-Aid SR(C) 07640280 and also by JMESC
Grant-in-Aid CR(A) 05302012.
$*)$

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remains valid under non-minor change of the basic setting. However, just for simplicity we adopt in this article the so-called original standard setting [11] in white
noise analysis or Hida calculus to state our results related to the PFM transform.

\S 2. Notation and Preliminaries
Let
be the Schwartz class space onandits dual space.
Then
is a Gelfand triple. We define the family of norms, where the operator $A=-d^{2}/dt^{2}+t^{2}+1$
given by $|\xi|_{p}=|A^{p}\xi|,$ $p>0,$
andis the-norm. Let
be the completion ofwith respect$|\cdot|_{p},p>0$, and we have
. We denote its dual space byto the norm
. Let be the standard Gaussian measure on
such
that
$\int_{s*}\exp(\sqrt{-1}\langle X, \xi\rangle)\mu(dx)=\exp(-\frac{1}{2}|\xi|^{2)}$ ,$S^{*}\equiv S’(\mathbb{R})$$\mathbb{R}$$S\overline{=}S(\mathbb{R})$$S(\mathbb{R})\subset L^{2}(\mathbb{R})\subset S’(\mathbb{R})$$\xi\in S(\mathbb{R})$$L^{2}(\mathbb{R})$$|\cdot|$$S(\mathbb{R})$$S_{p}\equiv S_{\mathrm{p}}(\mathbb{R})$$S_{p}^{*}\equiv S_{p}’(\mathbb{R})$$S_{p}(\mathbb{R})\subset L^{2}(\mathbb{R})\subset S_{p}’(\mathbb{R})$$S’(\mathbb{R})$

$\mu$

.
denotes the Hilbert space of complex-valued -square
for any. The Wiener-It\^o decomposition theorem
integrable functionals with norm
, i.e.,
gives the unique representation of in$(L^{2})$$\xi\in S(\mathbb{R})$$\mu$$||\cdot||$$(L^{2})$$\varphi$(1)$\varphi=\sum_{n=0}^{\infty}I(f_{n}n)$,$f_{n}\in\hat{L}_{\mathbb{C}}^{2}(\mathbb{R}^{n})$,the space of
where
denotes the multiple Wiener integral of order and. The second quantization operator
symmetric complex valued -functions on,
as follows: foris densely defined on$n$$I_{n}$$L^{2}$$(L^{2})$$\Gamma(A)$(2)$\hat{L}_{\mathbb{C}}^{2}(\mathbb{R}^{n})$$\mathbb{R}^{n}$$\varphi=\sum_{n=0}^{\infty}In(fn)\in \mathrm{D}\mathrm{o}\mathrm{m}(\Gamma(A))$$\Gamma(A)\varphi=\sum_{n=0}^{\infty}I(n)A^{\otimes n}f_{n}$.

and let $(S)_{p}\equiv\{\varphi\in(L^{2});||\varphi||_{p}<\infty\}$ and, defineFor. Let $(S)$ be the projective limit ofis denoted by
the dual space of. It is called a space of test white noise functionals. The elements inof $(S)$ are called generalized white noise functionals or Hida
the dual space
distributions. In fact, $(S)\subset(L^{2})\subset$ (S)*is a Gelfand triple [11]. For convention all(resp.
mean the canonical bilinear for.ms on, resp.
dual pairings$(S)^{*}\mathrm{x}(S))$ unless otherwise stated.
The -transform of $\Phi\in(S)^{*}$ is a function on defined by$||\varphi||_{p}=||\Gamma(A)p\varphi||$$p\in \mathbb{N}$$(S)_{p}$$(S)_{p}^{*}$$\{(S)_{p};p\in \mathbb{N}\}$$(S)^{*}$$\langle\cdot, \cdot\rangle$$S^{*}\cross S$$\langle\langle\cdot, \cdot\rangle\rangle$$S$$\mathrm{S}$(3)$(S\Phi)(\xi):=\langle\langle\Phi, :\exp\langle\cdot, \xi\rangle:\rangle\rangle$,$\xi\in S(\mathbb{R})$,Then note that a mapping :$\eta\in S$ . A complex valued function $F$$(S\Phi)(Z\xi+\eta)$ is entire holomorphic for any
on is called a -functional if and only if it is ray entire on and if there exist$C_{2}>0$ , andso that the estimate
constants

where :$\mathbb{C}\ni z-\neq$$\exp\langle\cdot, \xi\rangle:\equiv\exp\langle\cdot, \xi\rangle\cdot\exp(-\frac{1}{2}|\xi|^{2}\mathrm{I}\cdot$$\xi,$

$S$

$U$

$S$

$p\in \mathbb{N}\cup\{0\}$$C_{1},$$|F(z\xi)|\leq C_{1}\exp(c2|z|^{2}|\xi|_{p)}2$

may hold for all[25]:$z\in \mathbb{C},$$\xi\in S$. We have the following Characterization Theorem

35

is a -functional. Conversely, if $F$ is a UTheorem 1. If $\Phi\in(S)^{*}$ , thensuch that $S\Phi=F$ holds.
functional, then there exists a unique element in$U$$S\Phi$$(S)^{*}$$\Phi$

Based upon the above characterization we are able to give rigorous definitions
to Fourier type transforms of infinite dimensions. The Kuo type Fourier transform$[16,17]$ of a generalized white noise functionalinis the generalized white
noise functional, -transformation of which is given by$(S)^{*}$$\Phi$$\mathcal{F}$$\mathrm{S}$(4)$S(\mathcal{F}\Phi)(\xi)=\langle\langle\Phi, \exp(-i\langle\cdot, \xi\rangle)\rangle\rangle$,$\xi\in S$.

Likewise, the Fourier-Mehler transformof a generalized white noiseis the generalized white noise functional, -transformation offunctional inwhich is given by$\mathcal{F}_{\theta}(\theta\in \mathbb{R})[18]$$(S)^{*}$$\Phi$(5)$\mathrm{S}$$S( \mathcal{F}_{\theta}\Phi)(\xi)=\langle\langle\Phi, \exp\{^{i}\mathrm{e}\theta\langle\cdot, \xi\rangle-\frac{1}{2}\mathrm{e}^{i\theta}\cos\theta|\xi|^{2}\}\rangle\rangle$,$\xi\in S$.

is a generalization of the Kuo type Fourier
The Fourier-Mehler transform, andis coincident with the Fourier transform
transform . Actually,. Hence we haveis the inverse Fourier transform. It is easy to see that$\mathcal{F}_{\theta},$$\mathcal{F}_{0}=Id$$\mathcal{F}$$\theta\in \mathbb{R}$$\mathcal{F}_{-\pi/2}$$\mathcal{F}^{-1}$$\mathcal{F}$$\mathcal{F}_{\pi/2}$$S( \mathcal{F}^{-}1\Phi)(\xi)=(S\Phi)(i\xi)\exp(-\frac{1}{2}|\xi|^{2)},$\S 3.

$\xi\in S$.

-Mehler Transform$\mathrm{P}\mathrm{s}\mathrm{e}\mathrm{u}\mathrm{d}\mathrm{o}-\mathrm{F}\mathrm{o}\mathrm{u}\mathrm{r}\mathrm{i}\mathrm{e}\mathrm{r}$We begin with introducing theanalysis.

form

given byif$\{\Psi_{\theta}, \theta\in \mathbb{R}\}$$\Psi_{\theta}$(6)-Mehler transform in white noise

is said to be the $Pseudo-F_{ouri}er-Mehler(PFM)$ trans, whose -functional isinto itself foris a mapping from

Definition 1.
$[\mathit{5},\mathit{6}]$$\mathrm{p}\mathrm{s}\mathrm{e}\mathrm{u}\mathrm{d}_{0^{- \mathrm{F}\mathrm{i}\mathrm{e}}}\mathrm{o}\mathrm{u}\mathrm{r}\mathrm{r}$$U$$\theta\in \mathbb{R}$$(.S)^{*}$$S(\Psi_{\theta}\Phi)(\xi)=F(e^{i\theta}\xi)\cdot\exp(ie\sin\theta|\xi|^{2})i\theta$,$\xi\in S$,or equivalently(7)$S( \Psi_{\theta}\Phi)(\xi)=\langle\langle\Phi, \exp(^{i\theta}e\langle\cdot, \xi\rangle-\frac{1}{2}|\xi|^{2})\rangle\rangle$, wherefor-functional of .
$\Phi\in(S)^{*}$

$U$

$S$

is the$S$,$\xi\in S$,-transform in white noise analysis and$F$denotes the$\Phi$are UandBy virtue of Theorem 1, the right hand sides inabove-mentioned. Therefore theinfunctionals, andexists for each
-Mehler transform is well-defined. Hence we have$\mathrm{E}\mathrm{q}.(6)$$\Psi_{\theta}\Phi$$\mathrm{p}_{\mathrm{S}\mathrm{e}\mathrm{u}}\mathrm{d}_{0^{- \mathrm{F}_{0}}}\mathrm{u}\mathrm{r}\mathrm{i}\mathrm{e}\mathrm{r}$$\Phi$$(S)^{*}$$\mathrm{E}\mathrm{q}.(7)$36

Proposition 2. The following properties hold:(i) $\Psi_{0}=Id;$ ( $Id$ denotes the identity operator.);(ii)
for any
.
any
(iii)
for
$\theta\in \mathbb{R}\backslash \{0\}$$\Psi_{\theta}\neq \mathcal{F}$$\theta\in \mathbb{R}\backslash \{0\}$$\Psi_{\theta}\neq \mathcal{F}_{\theta}$

Proof. As to (i), it is easy to see that $S(\Psi_{0}\Phi)(\xi)=S\Phi(\xi)=F(\xi)$ . The characterization theorem allows the equality $\Psi_{0}=Id$ . (iii) is obvious from definitions.
, it follows clearly from (iii) thatnever coincides
and
Since
$\theta=0$.
exceptwithfor any$\mathcal{F}_{0}=Id$$\mathcal{F}$$\mathcal{F}_{-\pi/2}=\mathcal{F}$$\theta\in \mathbb{R}$$\Psi_{\theta}$$\square$Proposition 3. The invese operator of the $P_{S}eudo-Fourier$-Mehler.
is given by
for$(\Psi_{\theta})^{-1}=\Psi_{-\theta}$

Proof. It is sufficient to show thatfor $\Phi\in(S)^{*}$ we get from the definition (6)$\Psi_{-\theta}\Psi_{\theta}=\Psi_{\theta}\Psi_{-\theta}=Id$(8)$tran\mathit{8}form\Psi_{\theta}$$\theta\in \mathbb{R}$. As a matter of fact,$S(\Psi_{-\theta}(\Psi_{\theta}\Phi))(\xi)=S(\Psi_{\theta}\Phi)(\mathrm{e}-i\theta\xi)\cdot\exp(-i\mathrm{e}-i\theta\sin\theta|\xi|2)$$=(S\Phi)(\mathrm{e}^{i\theta}(\mathrm{e}-i\theta\xi))\cdot\exp(i\mathrm{e}^{i\theta i}\sin\theta|\mathrm{e}-\theta\xi|2)\cdot\exp(_{-i\mathrm{s}}\mathrm{e}^{-i}\mathrm{i}\mathrm{n}\theta|\xi|\theta 2)$$=(S\Phi)(\xi)$. exp(O) $=S(.Id\cdot\Phi)(\xi)$ ,$\xi\in S$,

because we used the relation$S(\Psi_{-}\theta\Phi)(\xi)=S\Phi(\mathrm{e}^{-i}\xi\theta)\cdot\exp(-i\mathrm{e}^{-i}\mathrm{s}\mathrm{i}\theta \mathrm{n}\theta|\xi|^{2})$. An application of the characterization
so as to obtain the second line of. As for the other part of the desired equalities,
gives
theorem toit goes almost similarly.
is like (see CorolNext let us consider what the image of the space $(S)$ underalso enjoys some interesting-Mehler transformlary 6 below). The(see Theorem 4 andwhitefunctionals
noise
properties on the product of Gaussian
Theorem 5).
$\mathrm{E}\mathrm{q}.(8)$$\Psi_{-\theta}\Psi_{\theta}=Id$$\mathrm{E}\mathrm{q}.(8)$$\square$$\Psi_{\theta}$$\mathrm{p}_{\mathrm{S}}\mathrm{e}\mathrm{u}\mathrm{d}\mathrm{o}-\mathrm{F}_{\mathrm{o}\mathrm{u}\mathrm{r}\mathrm{i}}\mathrm{e}\mathrm{r}$$\Psi_{\theta}$

be a Gaussian white noise functional, $i.e,.g_{\mathrm{c}}(\cdot):=N\exp(-|\cdot$
Theorem 4. Let$|^{2}/2c)$ with renormalization $N$ and$c\neq 0,$ $-1$ .
the following
For
hold:
$g_{c}$$\theta\in \mathbb{R}$$c\in \mathbb{C},$$equalitie\mathit{8}$$(i)\Psi\theta\Phi$:$g_{C(\theta)}=\Gamma(e^{i\theta}Id)\Phi$(ii) for any$p\in \mathbb{R}$

,

.

,

$||\Psi_{\theta}\Phi$$\forall\Phi\in(S)^{*}f$:$g_{\mathrm{C}(}\theta$) $||_{p}=||\Phi||_{p}$ ,where : denotes the Wick product ($e.g$ .
by $c(\theta)=-(2^{-1}ie^{-i\theta}\csc\theta+1)$ .
Proof. Noting that the
we readily obtain(9)$S(\Psi_{\theta}\Phi :$\mathrm{U}$$[\mathit{1}\mathit{1}$-functional of.

$\forall\Phi\in(S)_{p_{f}}$, p.101]) and the parameter$g_{c}$$c(\theta)$

is given

is given by $\exp(-2^{-1}(1+c)^{-1}|\xi|^{2})$ ,g_{c}(\theta))(\xi)=S(\Psi\theta\Phi)(\xi)\cdot(Sg_{C}(\theta))(\xi)$$=S\Phi(\mathrm{e}^{i\theta-}\xi)\cdot--(\theta, \xi)$,$\xi\in S$,

37

because we employed

and put$\mathrm{E}\mathrm{q}.(6)$$—( \theta, \xi):=\exp(i\mathrm{e}^{i\theta}\sin\theta|\xi|^{2}-\frac{1}{2(1+c(\theta))}|\xi|^{2})$Then we cannot find any(9)$\theta\in \mathbb{R}$.

such that$=S \Phi(\xi)=\exp(-\frac{1}{2}|\xi|^{2)}\cdot\langle\langle\Phi, \mathrm{e}^{\{\cdot,\xi\rangle}\rangle\rangle$

for any $\Phi\in(S)^{*}$ . However, when:
may hold, which implies that), then(the symmetric space:,:, so that, we easily get fromis given byits -functional
operatorquantization
definition of the second$\Psi_{\theta}\Phi$$\Phi(x)=\sum_{n=0}^{\infty}$$x^{\otimes n}$$f_{n}\rangle,f_{n}\in\hat{S}_{-p}(\mathbb{R}^{n})$$\langle$$S\Phi(\xi)$$\mathrm{U}$$g_{C(\theta)}\neq\Phi$$S_{-p}(\mathbb{R}^{n})$$\sum_{n=0}^{\infty}\langle\xi^{\otimes n}, f_{n}\rangle$$\Gamma$r.h.s. of (9).

$= \sum_{n=0}^{\infty}\langle(\mathrm{e}^{i})^{n}\theta\xi\otimes n, f_{n}\rangle\cdot---(\theta, \xi)=s(\Gamma(\mathrm{e}Id)i\theta\Phi)(\xi)\cdot---(\theta, \xi)$holds, thenHence, ifwith the characterization theorem that$2i(1+c(\theta))\mathrm{e}^{i\theta}\sin\theta=1$$\Psi_{\theta}\Phi$:$\mathrm{c}\mathrm{l}\mathrm{e}\mathrm{a}\mathrm{r}\mathrm{l}\mathrm{y}^{-}--(\theta, \xi)$$g_{C}(\theta)=\Gamma(\mathrm{e}^{i\theta}Id)\Phi$proves to be 1, suggesting.

Moreover, it is easy to see that$||\Psi_{\theta}\Phi$:$g_{c}(\theta)||_{p}=||\Gamma(\mathrm{e}^{i\theta}Id)\Phi||p=||\Phi||_{p}$.
holds for any
If we take the assertion obtained in Theorem 4 into account, then the following) can be(i.e.
questions will arise naturally: whether the PFM transformed
represented by the Wick product of something like a transformed and a Gaussian
white noise functional ; furthermore, if so, what is the parameter $c=c(\theta)$ then?
, a simple:,:
First of all, on the assumption that$\xi\inS$
computation gives, for$p\in \mathbb{R}$$\square$$\Phi$$\Psi_{\theta}\Phi$$\Phi$$g_{c}$$\Phi(x)=\sum_{n=0}^{\infty}$(10)$\langle$$x^{\otimes n}$$f_{n}\rangle\in(S)^{*}$$S(\Psi_{\theta}\Phi)(\xi)=S\Phi(\mathrm{e}^{i\theta}\xi)\cdot\exp(i\mathrm{e}^{i\theta}\sin\theta|\xi|^{2})$$=S(\Gamma(\mathrm{e}^{i\theta}Id)\Phi)(\xi)\cdot\exp(i\mathrm{e}^{i}\sin\theta|\xi|^{2)}\theta$

We know fromeven for any:$\mathrm{E}\mathrm{q}.(10)$$\Phi$$g_{K(\theta)}$(11)$s(\Phi :that there is no possibility that, because$\Psi_{\theta}\Phi$.

may coincide with

$K(\theta),\theta\in \mathbb{R}$g_{K(\theta}))(\xi)=s\Phi(\xi)\cdot(sgK(\theta))(\xi)=S\Phi(\xi)\cdot\Lambda(K(\theta), \xi)$with$\Lambda(r, \xi):=\exp\{-\frac{1}{2(1+r)}|\xi|^{2}\}$.

38

On the other hand, since the -transform of$\mathrm{S}$:$\Gamma(\mathrm{e}^{i\theta})\Phi$$g_{K(\theta)}$

is given by,$S(\Gamma(\mathrm{e}^{i\theta}Id)\Phi)(\xi)\cdot\Lambda(K(\theta), \xi)$

it is true from (10) that$\Psi_{\theta}\Phi=\Gamma(\mathrm{e}i\theta Id)\Phi$may possibly hold for $\Phi\in(S)^{*},$:$g_{K}(\theta)$

as far as$\theta\in \mathbb{R}$$2\dot{\iota}(1+K(\theta))\mathrm{e}^{i\theta}\sin\theta+1=0$

is satisfied. Let us next consider the evaluation of the term
to the-norm. We need to determine the parameterfrom the relation between
and:calculation in (10) we readily obtain

relative, which comes. By a similar$\Psi_{\theta}\Phi(\Phi\in(S)_{p})$$(S)_{p}$$(p\in \mathbb{R})$$\Gamma(\mathrm{e}^{i\theta})(\Phi :$\Gamma(\mathrm{e}^{i\theta})\Phi$$g_{K(\theta)}$(12)$S(\Gamma(\mathrm{e}^{i}I\theta d)(\Phi :g_{A(\theta)}))(\xi)=S(\Phi :$A(\theta)$

g_{A(\theta)})$g_{A(\theta)})(\mathrm{e}^{i\theta}\xi)$$=(s\Phi)(\mathrm{e}^{i}\xi\theta)\cdot\Lambda(A(\theta), \mathrm{e}i\theta)$$=(S \Phi)(\mathrm{e}^{i}\xi\theta)\cdot\exp\{-\frac{\mathrm{e}^{2i\theta}}{2(1+A(\theta))}|\xi|^{2}\}$by making use of Eq.(ll). A comparison of (12) with
provides with$\Gamma(\mathrm{e}^{i\theta}Id)\Phi$as far as:$g_{K(\theta)}=\Gamma(\mathrm{e}^{i\theta}Id)(\Phi :$A(\theta)=2^{-1}i\mathrm{e}^{-i\theta}\csc\theta-1$:, and anyg_{A(\theta)})$$gK(\theta)||_{p}$$=||\Gamma(\mathrm{e}^{i\theta}Id)(\Phi :$\Phi\in(S)_{p},p\in \mathbb{R}$$S(\Gamma(\mathrm{e})i\theta\Phi)(\xi).\Lambda(K(\theta), \xi)$. It therefore follows that$||\Psi_{\theta}\Phi||_{p}=||\Gamma(\mathrm{e}^{i\theta}Id)\Phi$for all,$\theta\in \mathbb{R}$gA(\theta))||p=||\Phi$:$gA(\theta)||_{p}$. Summing up, we thus obtain

Theorem 5. The following equalities hold for any(i) if $K(\theta)=2^{-1}ie^{-i\theta}\csc\theta-1$ , then$\Psi_{\theta}\Phi=\Gamma(e^{i\theta}Id)\Phi$:$g_{K}(\theta)$,$\theta\in \mathbb{R}$:$\Phi\in(S)^{*};$(ii) if $A(\theta)=2^{-1}ie^{-i\theta}\csc\theta-1$ , then$||\Psi_{\theta}\Phi||_{\mathrm{P}}=||\Phi$for all$p\in \mathbb{R}$:$gA(\theta)||_{p}$,$\Phi\in(S)_{p}$.

under the-Mehler transform.
Let us think of the image of$g_{d}=1$
holds with $c+d=-2$ . So we have
It is easily checked that:$\varphi\in(S)$$\mathrm{p}\mathrm{s}\mathrm{e}\mathrm{u}\mathrm{d}\mathrm{o}- \mathrm{F}_{\mathrm{o}\mathrm{u}\mathrm{r}\mathrm{i}}\mathrm{e}\mathrm{r}$$g_{c}$(13)$g_{C(}\theta):g_{K(\theta)}=1$

From (ii) of Theorem 4, immediately,$\Psi_{\theta}\varphi$$\varphi\in(S)$:.

if and only if$g_{c}(\theta)\in(S)$,so that, it is equivalent to$\Psi_{\theta\varphi:g_{C}(\theta}):g_{K(\theta)}\in(S):g_{K}(\theta)$,:
denotes the whole space of elements
where $(S)$ :by
obviousthat
virtueofConsequently, it is:.
we obtain$g_{K(\theta)}$$\varphi$$\Psi_{\theta}\varphi\in(S)$$g_{K(\theta)},$$g_{K(\theta)}$for $\varphi\in(S)$ .
. Therefore$\mathrm{E}\mathrm{q}.(13)$39Corollary 6. For$\theta\in \mathbb{R}$,${\rm Im}\Psi_{\theta}(S)=(S):gK(\theta)\equiv\{\varphi :gK(\theta);\varphi\in(S)\}$,

where $K(\theta)=2^{-1}ie^{-i\theta}\csc\theta-1$ .
Remark 2. The results in Theorem 4 and Theorem 5 are quite similar to those$\Phi\in(S)_{p}$ ,of the Fourier-Mehler transform. In fact, for$p\in \mathbb{R},$$||(\mathcal{F}_{\theta}\Phi)$:$g_{c_{1}(\theta)}||_{p}=||\Phi||_{p}$and$||\mathcal{F}_{\theta}\Phi||_{p}=||\Phi$:$g_{c_{2}(\theta)}||_{p}$hold with $c_{1}(\theta)=-i\cot\theta-2$ , and $c_{2}(\theta)=i\cot\theta-2$ (e.g. [11,\S 9.H]).
is given by
Remark 3. The image of $(S)$ under the Fourier-Mehler transform$(S)$ :
, while that of $(S)$ under the Fourier transform
coincides with the$\mathcal{F}_{\theta}$$\mathcal{F}$$g_{i\cot\theta}$

space
$(S)$

where

$\tilde{\delta}_{0}$:is the delta function at$\tilde{\delta}_{0}\equiv\{\varphi :$0$\tilde{\delta}_{0;}\varphi\in(S)\}$,

and$\lim_{carrow 0}g_{C}=\overline{\delta}0$(e.g. [11, Chapter 9]).

\S 4. Infinitesimal Generators
First of all, for all$\theta\in S$

we define$\varphi_{\xi}(_{X)}$$:= \sum_{n=0}^{\infty}\frac{1}{n!}\langle:x^{\otimes n}:, \xi^{\otimes n}\rangle$

with $x\in S^{*},$ $\xi\in S$ . We call it an exponential vector. Then
operator on $(S)$ defined by
(14)
Let

denote the distribution in$(S\otimes S)^{*}$

is an

.

$(G_{\theta}\varphi\xi)(x):=\varphi \mathrm{e}^{i\theta}\xi(X)\cdot\exp(i\mathrm{e}i\theta 2\sin\theta|\xi|)$$\tau$$\{G_{\theta}, \theta\in \mathbb{R}\}$

given by$\langle\tau, \xi\otimes\eta\rangle=\langle\xi, \eta\rangle$,$\xi,$$\eta\in S$.

Note that it can be expressed as,$\tau=\int_{\mathbb{R}}\delta_{t}\otimes\delta_{t}dt=\sum_{j=0}ej\otimes e_{j}\infty\in(S\otimes S)*$

where$\{e_{n}\}$denotes a complete orthonormal basis for$L^{2}(\mathbb{R})$. Moreover we have$\tau^{\otimes n}=\int_{\mathbb{R}^{n}}\delta_{t_{1}}\otimes\delta_{t_{1}}\otimes\cdots\otimes\delta_{t_{n}}\otimes\delta_{t_{n}}dt_{1}\cdots dtn$.

The following is an easy exercise. The next lemma provides with a general expression for elements of general form in $(S)$ .

40

Lemma 7. When, then$S(\mathbb{R}^{n}))$$\varphi(x)=\sum_{n=0}^{\infty}\langle:X^{\otimes n} :, f_{n}\rangle\in(S)$$G_{\theta}\varphi i_{\mathit{8}}$with$f_{n}\in\hat{S}(\mathbb{R}^{n})$,(the symmetricgiven by$(c_{\theta\varphi})(X)= \sum_{n=0}\langle:x\infty\otimes n:, g_{n}\rangle$,

and$g_{n} \equiv g_{n}(\varphi)=\sum^{\infty}\frac{(n+2m)!}{n!m!}m=0(i\sin\theta)^{m}ei(n+m)\theta\otimes m_{*f_{2}}\tau m+n$where for the element $f_{2m+n}$ in
following integral expression$\hat{s}(\mathbb{R}^{2m+n})$

the term’

actually has the$\tau^{\otimes m}*f_{2m+n}$$(\tau^{\otimes m_{*fn}}2m+)(t_{1}, \cdots, t_{n})$$= \int_{\mathbb{R}^{m}}f_{2n}m+(S_{1}, s_{1}, \cdots, s_{m}, s_{m}, t_{1}, \cdots, t_{n})dS_{1}\cdots dS_{m}$.

On this account, we obtain immediately
Proposition 8. The $Pseud_{\mathit{0}-}Fourier$-Mehler transform$i.e.$ ,adjoint operator of$\{\Psi_{\theta;}\theta\in \mathbb{R}\}$is given by the$\{G_{\theta;}\theta\in \mathbb{R}\},$$\Psi_{\theta}=G_{\theta}^{*}$

holds in operator equality sense for all$\theta\in \mathbb{R}$.

The next proposition gives an explicit action of the PFM transform
for thegeneralized white noise functionals of general form. It is due to a direct computation.
$\Psi_{\theta}$

Proposition 9. For $\Phi\in(S)^{*}$ given as$\Phi(x)=\sum_{n=0}^{\infty}$$\langle:x^{\otimes n}:,F_{n}\rangle$$\Psi_{\theta}\Phi(x)=n\sum^{\infty}\langle:X^{\otimes n}:,\sum_{+l2m=n}a(l=0’\otimes m, \theta)\cdot F_{l^{\wedge}}\tau\otimes m\rangle$, it holds that,

where the constant $a(l, m, \theta)$ is given by$a(l, m, \theta)=\frac{1}{m!}e^{i(lm)}+\theta(i\sin\theta)^{m}$.

Remark 4. Similar results for Fourier-Mehler transform as the above can befound in [23]. For the proof of Proposition 9, it is almost the same as those given
in [23].
It follows from Proposition 3 that the-Mehler transform
isinjective and surjective. Moreover, it is easy to check thatis a strongly continuous operator from
into itself, when we take Lemma 7 and Proposition 8 into
consideration. Thus we have the following theorem.
$\mathrm{p}_{\mathrm{S}\mathrm{e}}\mathrm{u}\mathrm{d}_{0}-\mathrm{F}_{0}\mathrm{u}\mathrm{r}\mathrm{i}\mathrm{e}\mathrm{r}$$\Psi_{\theta}$$\Psi_{\theta}$$(S)^{*}$

41-Mehler transform
Theorem 10 [5]. The
continuous linear operator.
bijective and$P_{Seu}d_{\mathit{0}-}Fourier$$\Psi_{\theta}$: $(S)^{*}arrow(S)^{*}$ is a$\mathit{8}trongly$

Theorem 11 [5]. The set
forms
continuous linear operator acting on the space$\{\Psi_{\theta;}\theta\in \mathbb{R}\}$

Proof. For $\Phi\in(S)^{*},$ $\xi\in S$ , and any
(15)

$a$

$\eta\in \mathbb{R}$$\theta,$

one parameter group of stronglyof Hida distributions.

$(S)^{*}$, from (7) of Definition 1 we have

$S( \Psi_{\theta+\eta}\Phi)(\xi)=\langle\langle\Phi, \exp\{\mathrm{e}^{i(\theta}+\eta)\langle\cdot, \xi\rangle-\frac{1}{2}|\xi|^{2}\}\rangle\rangle$.

While, from (6)(16)$S(\Psi_{\theta}(\Psi_{\eta}))(\xi)=S(\Psi_{\eta}\Phi)(\mathrm{e}^{i\theta}\xi)\cdot\exp(i\mathrm{e}\sin\theta|\xi|i\theta 2)$$=F(\mathrm{e}^{i\eta}(\mathrm{e}^{i}\theta\xi)),$ $\exp(i\mathrm{e}\sin\eta|\mathrm{e}^{i}\xi\theta|2i\eta)\cdot\exp(i\mathrm{e}^{i\theta}\sin\theta|\xi|^{2})$$=\mathrm{e}^{-\frac{1}{2}}|\mathrm{e}^{i(e+}\eta)\xi|^{2}\langle\langle\Phi, \mathrm{e}^{\langle\cdot,\mathrm{e}^{(\theta+}\xi\rangle}i\eta)\rangle\rangle\cdot\exp\{i\mathrm{e}^{i\theta}(\mathrm{e}^{i(\theta+\eta}\mathrm{s}\mathrm{i})\mathrm{n}\eta+\sin\theta)|\xi|^{2}\}$$= \langle\langle\Phi, \exp\{\mathrm{e}^{i}\langle(\theta+\eta)., \xi\rangle-\frac{1}{2}|\xi|^{2}\}\rangle\rangle$

with the$\mathrm{U}$-functional $F$ of$\Phi$,. By comparing (15) with (16), we get$S(\Psi_{\theta}+\eta\Phi)(\xi)=S(\Psi\theta\Psi\eta\Phi)(\xi)$.

Consequently, the characterization theorem leads to$\Psi_{\theta+\eta\eta}\Phi=\Psi_{\theta}\cdot\Psi\Phi$,$\Phi\in(S)^{*}$,

which completes the proof.
We are now in a position to state one of the principal results in this paper. Thisis a very important property of the Pseudo-Fourier-Mehler transform, especially on
an applicational basis.
$\square$$i(N+\Delta_{G}^{*})$,$\Delta_{G}$.

is given by
Theorem 12 [5]. The infinitesimal generator of
Laplacianis the adjoint of thewhere $N$ is the number operator and$\{\Psi_{\theta;}\theta\in \mathbb{R}\}$$Gro\mathit{8}S$$\Delta_{G}^{*}$

Remark 5. It is well known that the infinitesimal generator of the Fourier-Mehlerhas, while the adjoint operator ofistransforms
as its infinitesimal generator (e.g. see [11]). The proof of Theorem 12 isalmost similsr to the above ones.
Proof of Theorem 12. First of all we set$\{\mathcal{F}_{\theta;}\theta\in \mathbb{R}\}$$\{\mathcal{F}_{\theta;}\theta\in \mathbb{R}\}$$iN+ \frac{i}{2}\Delta_{G}^{*}$$iN+ \frac{i}{2}\Delta_{G}$$F_{\theta}(\xi):=S(\Psi_{\theta}\Phi)(\xi)$and$F_{0}(\xi):=s(\Phi)(\xi)$

for $\Phi\in(S)^{*},$ $\xi\in S$ , paying attention to (i) of Proposition 2. From (6) we have
. Since is Fr\’echet differentiable, the functional$F_{\theta}(\xi)=F_{0}(\mathrm{e}^{i\theta})\cdot\exp[i\mathrm{e}^{i\theta}\sin\theta|\xi|^{2}]$$F_{0}$

42$F_{\theta}(\xi)$is differentiable in(17)$\theta$

as well, and it is easy to check that$\lim_{\thetaarrow 0}\frac{1}{\theta}\{F\theta(\xi)-F0(\xi)\}$$=\langle F_{0}’(\mathrm{e}^{i}\xi\theta), i\mathrm{e}i\theta\rangle\cdot\exp(i\mathrm{e}\theta i\mathrm{i}\mathrm{s}\mathrm{n}\theta|\xi|2)|_{\theta=0}$$+F_{0}( \mathrm{e}^{i\theta}\xi)\cdot\frac{d}{dt}\exp(i\mathrm{e}^{i\theta}\sin\theta|\xi|2)|_{\theta=0}$$=i\langle F’(\xi), \xi\rangle+i|\xi|^{2}\cdot F(\xi)$.

While, we can easily check that the -functionalso thatsatisfies the uniform bounded criterion:$\theta^{-1}\cdot\{F_{\theta}(\xi)-F_{0}(\xi)\},$$\mathrm{U}$$\theta\in \mathbb{R}$$\exists C_{0}>0$$\sup_{z\in \mathbb{C}}|\frac{1}{\theta}\{\tilde{F}_{\theta}(Z\xi)-\tilde{F}_{0}(_{Z}\xi)\}|\leq C_{0}\exp(c_{1}R^{C}2|\xi|_{p}^{2})$$|z|=R$denotes an entire
holds for all $R>0$ , all $\xi\in S$ with $c_{1}>0,$ $c_{2}>0$ , where
theorem [25] (seecriterion
convergencethestronganalytic extension of . Hence,
also [11, Chapter 4] allows convergence of$\tilde{F}_{*}$

$F$

$)$

$S^{-1}( \frac{1}{\theta}\{F_{\theta}(\cdot)-F_{0}(\cdot)\})(X)=\frac{1}{\theta}\{\Psi_{\theta}\Phi(_{X})-\Phi(x)\}$in$(S)^{*}$

as$\theta$tends to zero. We need the following two lemmas.

Lemma 13. (cf. [11, Theorem 6.11,p.196]) Let $F(\xi)=s\Phi(\xi),\xi\in S$ for $\Phi\in(S)^{*}$ .
Then(i) $F$ is Fr\’echet differentiable;$\xi\in S$ ;(ii) the -transform of $N\Phi(x)$ is given by
where $N$ is the number operator.
$S$$\langle F’(\xi), \xi\rangle,$

Lemma 14. (cf. [11, Theorem 6.20, p.206]) For any$\xi\in S$ .
is given by$\Delta_{G}^{*}\Phi(x)$$\Phi$in$(S)^{*}$

, the

$S$

-transform of$|\xi|^{2}S\Phi(\xi),$

We may deduce at once that
(18)

$S(N\Phi+\Delta^{*}G)\Phi(\xi)=\langle F’(\xi), \xi\rangle+|\xi|^{2}F(\xi)$,$\xi\in S$,

with simple applications of Lemma 13 and Lemma 14. Moreover, it is easily verified
from (17) and (18) together with the above-mentioned convergence result that$\lim_{\thetaarrow 0}\frac{1}{\theta}(\Psi_{\theta}-Id)\Phi(x)=\theta\lim s^{-1}arrow 0(\frac{1}{\theta}\{F_{\theta}(\cdot)-F_{0}(\cdot)\})(X)$$=S^{-1}(i\langle F’(\xi), \xi\rangle+i|\xi|2.

$=i(N+\Delta^{*}G)\Phi(x)$which completes the proof.

$\square$,F(\xi))$in$(S)^{*}$,

43\S 5. Application of PFM Transform
The purpose of this section is to show a typical example of application of theto the Cauchy problem.
Pseudo-Fourier-Mehler transform
Example 6. (A simple application of the PFM transform) Let us consider thefollowing abstract Cauchy problem on the white noise space:$\Psi_{\theta}$(19)$\frac{\partial u(t,X)}{\partial t}=iNu(t, X)+\varphi(x)$$u(0, \cdot)=f(\cdot)\in(S)$,,with $t>0$ , where $N$ denotes the number operator. One of the most remarkable
benefits of white noise analysis consists in its application to differential equation
theory and how to solve the problem (cf. [1], [2,3], [4,8]). Especially in $[4,8]$ , byresorting to the analogy in the finite dimensional cases we have applied the infinite
dimensional Kuo type Fourier transform to the Cauchy problem for heat equation
type with Gross Laplacian, and have succeeded in derivation of the general solution
and also in direct verification for existence and uniqueness of the solution. On thisaccount, we think of using the Fourier transform to the aforementioned problem.
Recall the formula:,(20)$\mathcal{F}(N\Phi)=N(\mathcal{F}\Phi)+\Delta_{G}^{*}(\mathcal{F}\Phi)$

We set $v(t, y)\equiv(\mathcal{F}u(t, \cdot))(y)$ for eachfor (19) so as to obtainform$t\in \mathbb{R}_{+}$

for all $\Phi\in(S)^{*}$ .

. We may employ the Fourier trans-$\mathcal{F}$

(21)

,

$\frac{\partial v(t,y)}{\partial t}=iNv(t, y)+i\Delta_{G}^{*}v(t, y)+\hat{\varphi}(y)$with$v(\mathrm{O}, y)=\hat{f}(y)$,. The operator part of the
because we made use of the formula (20) and set
Fourier transformed problem (21) is exactly equivalent to the infinitesimal generator
of PFM transform with parameter (see Theorem 12). Hence, the semigroup theory
in functional equation theory allows immediately the following explicit exression of
the solution in question:$\hat{F}=\mathcal{F}F$

$t$

(22)

$v(t, y)= \Psi t\hat{f}(y)+\int_{0}^{t}\Psi_{t-S}\hat{\varphi}(y)dS$.

We can show the existence and uniqueness of the solution by applying Theorem 4. In thatand Theorem 5 to (22) under a certain condition on the initial data
one.
type
Bochnerasinterpretedcase the integral term appearing in (22) should be
So much for the Cauchy problem, because this is not our main topic in this article.
We shall go back to the PFM transform and proceed further in the next section.
$\varphi,$

$f$

44

\S 6. Intertwining Properties
In this section we shall investigate some intertwining properties between the Pseudoand other typical operators in white noise analysis,
Fourier-Mehler transform
such as G\^ateaux differential, the adjoint of G\^ateaux differential, Hida differential
operator, and Kubo operator (the adjoint of Hida differential), etc. Furthermore,
we shall introduce the characterization theorem for PFM transforms, which is oneof our main results in this paper.
in the direction $y\in S^{*}$ .
We begin with definition of the G\^ateaux differential
$(S)$
$y\in
S^{*}$:,, we:
fixed, for the element ingiven byFor
put$\Psi_{\theta}$$D_{y}$$\varphi(x)=\sum_{n=0}^{\infty}$$\varphi$(23)$D_{y} \varphi(x)=\lim_{\thetaarrow 0}\frac{\varphi(x+\theta y)-\varphi(X)}{\theta}$,$x\in S^{*}$$x^{\otimes n}$$\langle$.

The limit existence in the right hand side of (23) is always guaranteed, andis actually given by(24)$D_{y} \varphi(x)=\sum_{n=0}^{\infty}n\langle:X\otimes(n-1):, y\otimes_{1}f_{n}\wedge\rangle$,$f_{n}\rangle$$x\in S^{*}$$D_{y}\varphi(x)$.

becomes a continuous linear operator from $(S)$ into itself. Since theIn fact,, adoption of
instead of does make sense inlies in
Dirac delta function
hand,
other
the
Hida differential operator
the
(23)
(24).
and
On
above
the
is originally proposed by T. Hida [9] and defined by$D_{y}$$S^{*}$$\delta_{t}$$\delta_{t}$$y$$\partial_{t}(=$$\partial/\partial x(t))$$\partial_{t}:=s^{-1}s\underline{\delta}$,$\xi\in S$$\delta\xi(t)$(cf. [15]; see also [7]). It is well known that the action of is equivalent to that ofon the dense domain [11] (or $[7],[14]$ ). So we can define$\partial_{t}$$D_{\delta_{t}}$$\partial_{t}=D_{\delta_{t}}$The Kubo operator$\partial_{t}^{*}[15]$,$t\in \mathbb{R}$.

is the adjoint of Hida differential$\partial_{t}$, defined by,$\langle\langle\partial_{t}^{*}\Phi, \varphi\rangle\rangle=\langle\langle\Phi, \partial_{t\varphi\rangle}\rangle$) can be considered as a(resp.
for $\Phi\in(S)^{*},$ $\varphi\in(S)$ . As a matter of fact,) into itself with respect to the weakcontinuous linear operator from $(S)$ (resp.
proves to be a continuous
differential
or strong topology. More precisely, the Hida$p\geq 0$ , while the Kubo operator turns
formapping from $(s)_{P+q}$ into$(S)_{-p}$$(S)_{-(p+q})$
as given above.
for the same pairinto
out to be the one fromis defined in the usual manner, andFor $\xi\in S,$ $\varphi\in(S)$ , the derivative: $(S)^{*}arrow(S)^{*}$ . Even for that, we shall henceforth use
there exists its extension
for brevity, as far as there is no confusion in the context. We
the same notation.
is the adjoint of
set $q_{\xi}:=i(D_{\xi}+D_{\xi}^{*})$ , where$\partial_{t}$$\partial_{t}^{*}$$(S)^{*}$$(S)_{q}$$q> \frac{1}{4},$$p,$$(D_{\xi\varphi})(X)$$\tilde{D}_{\xi}$$D_{\xi}$$D_{\xi}^{*}$$D_{\xi}$

$q$

45

Lemma 15. For each$\theta\in \mathbb{R},$$t\in \mathbb{R}$,$\Psi_{\theta}(\partial_{t}^{*}\Phi)=e^{i\theta*}\partial_{t}(\Psi\theta\Phi)$holds for all $\Phi\in(S)^{*}$ .

Proof. First of all, note that $S(\partial_{t}*\Phi)(\xi)=\xi(t).S(\Phi)(\xi)$ . So, for the generalized$x\in$:,:
white noise functional $\Phi\in(S)^{*}$ given in the form
we readily get$\Phi(x)=\sum_{n=0}^{\infty}$$x^{\otimes n}$$\langle$$f_{n}\rangle,$$S^{*}$(25)$S( \Psi_{\theta}(\partial_{t}^{*}\Phi))(\xi)=\mathrm{e}^{i\theta}\xi(t)\cdot\sum_{=n0}^{\infty}\langle f_{n}, \mathrm{e}^{i\theta n}n\xi^{\otimes}\rangle\cdot\exp(i\mathrm{e}^{i\theta}\sin\theta|\xi|^{2})$.

While we establish(26)$S(\Psi_{\theta}(\partial_{t}^{*}\Phi))(\xi)=\mathrm{e}si\theta(\partial_{t}^{*}(\Psi\theta\Phi))(\xi)$by applying (25), because we made use of the relation$S(\partial_{t}^{*}(\Psi_{\theta}\Phi))(\xi)=\xi(t)\cdot(S\Phi)(\mathrm{e}^{i\theta}\xi)\cdot\exp(i\mathrm{e}\sin i\theta\theta|\xi|^{2})$.

An application of the Potthoff-Streit characterization theorem (Theorem 1) to (26)leads to the required equality in Hida distribution sense.
$\square$Proposition 16. For each(i)(ii)
hold for all $\Phi\in(S)^{*}$ .

$\theta\in \mathbb{R},$$t\in \mathbb{R}$;$\Psi_{\theta}(\partial_{t}\Phi)=e^{-i\theta}\partial_{t}(\Psi_{\theta}\Phi)-2i\sin\theta\partial*(t\Phi\Psi_{\theta})$$\Psi_{\theta}(x(t)\Phi)=e^{-i\theta}X(t)(\Psi_{\theta}\Phi)_{f}$.

Remark 7. The assertion (i) of Proposition 16 follows from a direct .omputation.
We have only to employ the following two rules:$\mathrm{c}$$S \partial_{t}(\cdot)=\frac{\delta}{\delta\xi(t)}S(\cdot)$,$\partial_{t}^{*}(\cdot)=s^{-}1\xi(t)S(\cdot)$.

The second assertion (ii) is also due to a simple computation together with thefirst assertion (i) and Lemma 15. Moreover, we need to apply the multiplication(e.g. [19]). Those proofs go almost similarly as inoperator:very
easy, hence omitted.
arethe proof of Lemma 15 andThe next proposition indicates some intertwining property between the PFM
transform and G\^ateaux differential operator.
$x(t)(\cdot)=(\partial_{t}+\partial_{t}^{*})(\cdot)$

Proposition 17. For each parameter(i)$\theta\in \mathbb{R},t\in \mathbb{R}$$e^{-i\theta}\tilde{D}_{\xi(\Phi)}\Psi_{\theta}=\Psi_{\theta}(\tilde{D}_{\xi}\Phi)+2i\sin\theta\cdot D^{*}(\xi\Psi\theta\Phi)$(ii)
hold for all generalized white noise functionals in;$\tilde{D}_{\xi}(\Psi_{\theta}\Phi)+D_{\xi}^{*}(\Psi\theta\Phi)=e^{i\theta}\Psi_{\theta}(\overline{\langle\cdot,\xi}\rangle\Phi)_{i}$$(S)^{*}$.

and its adjoint
Proof. It is interesting to note that G\^ateaux differential
noise analysis
inwhite
expressions
enjoy the integral kernel operator theoretical(see the next section; or [11,12], [23]). Namely,$D_{\xi}$(27)$\tilde{D}_{\xi}:=(\int_{\mathbb{R}}\xi(t)\partial tdt)^{\sim}$,

and$D_{\xi}^{*}:= \int_{\mathbb{R}}\xi(t)\partial_{t}*dt$,$D_{\xi}^{*}$$\forall\xi\in S$.

46

Let $\Delta=\{t_{k}\}$ be a proper finite partition of the parameter space, anddenotes$1\leqk\leqm$
the maximum of increment
over. The assertion (i) yields from (i)of Proposition 16. In fact, by linearity of the PFM transform we get$t$$|\Delta|$$\triangle t_{k}$(28)$\sum_{k=1}^{m}\triangle t_{k}\xi(t_{k})\cdot\Psi_{\theta(\partial_{t_{k}}}\Phi)=\Psi_{\theta}(_{k}\sum_{=1}^{m}\xi(tk)\partial_{t_{k}}\triangle t_{k}\cdot\Phi)$,for. Consider the same type finite summation for the other terms in (i) of
Proposition 16. By taking the limit $marrow\infty$ and by continuity of(Theorem 10),
we can obtain the desired result with consideration of.
As to (ii), note first that we can have the expression$\forall\xi\in S$$\Psi_{\theta}$$\mathrm{E}\mathrm{q}.(27)$(29)$\tilde{q}_{\xi}=i\overline{\langle x,\xi\rangle}=(i\int_{\mathbb{R}}x(t)\xi(t)dt)^{\sim}$,by virtue of the multiplication operator $x(t)(\cdot)$ (cf. Remark 7). With (ii) of Proposition 16, we may take advantage of continuity ofand (29) to deduce that$\Psi_{\theta}$$\mathrm{e}^{-i\theta}(D_{\xi}+D_{\xi}^{*})(\Psi_{\theta}\Phi)=\mathrm{e}^{-i\theta}(\int_{\mathbb{R}}x(t)\xi(t)dt)(\Psi_{\theta}\Phi)$$= \lim_{marrow\infty}\Psi_{\theta}(_{k=0}\sum^{m}\triangle tk\xi(t_{k})x(t_{k})\cdot\Phi)$$=\Psi_{\theta}(\langle_{X}, \xi\rangle\cdot\Phi)$

by passage to the limit.
The following theorem gives the characterization fortransforms, which is one of our main results in this paper.
$|\triangle|arrow 0$$\square$-Mehler$\mathrm{p}\mathrm{s}\mathrm{e}\mathrm{u}\mathrm{d}\mathrm{o}- \mathrm{F}_{\mathrm{o}\mathrm{u}\mathrm{r}\mathrm{i}}\mathrm{e}\mathrm{r}$$\{\Psi_{\theta;}\theta\in \mathbb{R}\}$

Theorem 18 [6]. The $Pseudo-F_{ouri}er$ -Mehler transform
following conditions:: $(S)^{*}arrow(S)^{*}$ is a continuous linear operator for$(Pl)\Psi_{\theta}$

satisfies the$\{\Psi_{\theta;}\theta\in \mathbb{R}\}$$f_{ora}ll\theta\in \mathbb{R}_{f}$.

$(P\mathit{2})\Psi_{\theta}(\tilde{D}_{\xi}\Phi)=e^{i\theta}\overline{D}_{\xi(\Phi}\Psi_{\theta})-2\sin\theta\cdot\tilde{q}\xi(\Psi\theta\Phi)_{j}$$(P\mathit{3})\Psi_{\theta}(\tilde{q}\epsilon\Phi)=e^{-}\tilde{q}_{\xi}(i\theta\Psi_{\theta}\Phi)f$.

where $\Phi\in(S)^{*},$ $\xi\in S.$ Conversely, if a continuous linear operator: $(S)^{*}arrow$thenis a constant multiple ofsatisfies the above conditions:$A_{\theta}$$(S)^{*}$$\Psi_{\theta}$$(P\mathit{1})\sim.(P\mathit{3})_{f}$$A_{\theta}$.

Proof. (P1) is obvious (Theorem 10).
(. (P3)) yields fromof Proposition 17. It is due to a simple computation. Conversely, suppose that theoperator
be compatible withand (P3). We need the following results.
$(\mathrm{P}2)$$\mathrm{r}\mathrm{e}\mathrm{s}_{\mathrm{P}}$$(\mathrm{i})(\mathrm{r}\mathrm{e}\mathrm{s}_{\mathrm{P}}.

(\mathrm{i}\mathrm{i}))$$(\mathrm{P}1),(\mathrm{P}2)$$A_{\theta}$

Lemma 19. We assume thatbe a continuous linear operator from
into
itself, satisfying the three conditions. Then the following relations
(i)
;
.
(ii)
;
(iii)
hold for all $\xi\in S,$.
$(S)^{*}$$A_{\theta}$$(P\mathit{1})\sim(P\mathit{3})$$(\Psi_{\theta}-1---\theta)D\xi=D\xi(\Psi_{\theta}-1_{-}-_{\theta})-$$(\Psi_{\theta}^{-1_{-}-}--_{\theta})q\epsilon=q\xi(\Psi 1_{-}-)_{\rangle}\theta-\theta$$(\Psi_{\theta}^{-1_{-}}--\theta)D_{\xi}*=D_{\xi}^{*}(\Psi_{\theta}-1--_{\theta}-)$$\theta\in \mathbb{R}$The proof will be given below. The next theorem is well known (e.g.
Theorem 3.6, p.267] or [23, Prop.5.7.6, p.148] .
$)$

[12,

47

Theorem 20. Let A be a continuous linear operator on, for any $\xi\in S$ ;(i)(ii)
, for any $\xi\in S$ .
Then the operator A is a scalar operator.

$(S)^{*}$, satisfying$\Lambda\tilde{q}_{\xi}=\tilde{q}_{\xi}\Lambda$$\Lambda D_{\xi}^{*}=D_{\xi}^{*}\Lambda$

Thus, by taking
of Lemma 19 into account, we may apply Theorem 20
forto obtain the assertion:
is a scalar operator.
$(\mathrm{i}\mathrm{i}),(\mathrm{i}\mathrm{i}\mathrm{i})$$\square$$\Psi_{\theta}^{-1}A_{\theta}$$A_{\theta}$

Proof of Lemma 19. Basically it is due to a direct computation. Each proof goessimilarly, so we shall show only (iii) below. For the other two we will give justrough instructions. First of all, note that we have only to consider
instead ofby virtue of Proposition 3. As to (i), it is sufficient to calculate it with (P2)for both and (P3) for the PFM transform. As for, simply (P3) for bothand$\Psi_{-\theta}$$\Psi_{\theta}^{-1}$$A_{\theta}$$(,\mathrm{i}\mathrm{i})$$\Psi_{\theta}$. As to (iii), for(30)$\forall\Phi\in(S)^{*},$$\forall\xi\in S$,$(\Psi_{\theta}^{-}1A_{\theta})D^{*}\Phi\xi=-i\Psi_{\theta}-1(A_{\theta q}\xi)\Phi-\Psi_{\theta}-1(A\theta D_{\xi})\Phi$$=-\mathrm{e}^{i\theta}(\Psi_{\theta}^{-1}q_{\xi})A_{\theta}\Phi-\mathrm{e}(i\theta\Psi D\xi)\theta^{-1}A\theta\Phi$because we used a relation(31)$D_{\xi}^{*}=-i\tilde{q}_{\xi}-\tilde{D}_{\xi}$in the first equality and also employedin the second one. An applicationofto the last expression in (30), together with (31) again, gives$(\mathrm{P}2),(\mathrm{P}3)$$(\mathrm{P}2),(\mathrm{P}3)$(30)$=-iq\xi(\Psi_{\theta}^{-1}A\theta)\Phi-D_{\xi(}\Psi_{\theta}^{-}1A_{\theta})\Phi$$=(-iq\xi-D_{\xi})(\Psi_{\theta}^{-1}A\theta)\Phi=D\xi*(\Psi^{-}1A\theta)\theta\Phi$which completes the proof.

$\square$,48\S 7. Fock Expansion.
denote the space of continuous linear operators from $(S)$ intoLet
is a symmtrized space ofwith respect to the first ,The space
and the second $m$ variables independently. By virtue of the symbol characterization
theorem for operators on white noise functionals [21](see also [23]), for the operator
in
there exists uniquely a kernel distribution
II lying insuch that the operator may have the Fock expansion:$(S)^{*}$$\mathcal{L}((S), (S)^{*})$$S’(\mathbb{R}^{lm}+)$$\hat{S}_{l,m}’(\mathbb{R}l+m)$$l$$\mathcal{L}((S), (S)^{*})$$\kappa_{l,m}$$\hat{S}_{l,m}’(\mathbb{R}^{l+}m)$$\Pi$$\square =\sum_{l,m=0}^{\infty}\Gamma \mathrm{I}_{l},(ml,m)\kappa$.

$\varphi\in(S)$ converges in $(S)^{*}[21]$ . Generally, each component
Moreover, the series
II $l,m$ of the Fock expansion has a formal integral expression:$\Pi\varphi,$$\Pi_{l,m}(\kappa)=\int_{\mathbb{R}^{l+m}}\kappa(S_{1}, \cdots, s_{l}, t_{1}, \cdots, t_{m})$.

$\partial^{*}\partial\cdots\partial_{t_{m}}dS_{1}\cdots$\partial_{s_{1}s_{\iota}t_{1}}^{*}..,$.
dSldt1\ldots dt_{m}$.

Remark 8. We call it an integral kernel operator with kernel distribution . The
theory of integral kernel operators and the general expansion theory in white noise
analysis were proposed and have been developed enthusiastically by N. Obata [2123] (see also [11]). Those topics are closely related to quantum stochastic calculus,
which has been greatly investigated in chief by Hudson, Meyer, and Parthasarathy.
.Parthasarathy:
More details on this topic will be found in, for instance, (i)An Introduction to Quantum Stochastic Calculus, Birkh\"auser, Basel, 1992; (ii)
.Meyer: Quantum Probability for Probabilists, Lecture Notes in Mathematics
Vol.1538, Springer-Verlag, Heidelberg, 1993.
We shall give below two typical examples of the integral kernel operators in white
noise analysis.
Example 9. (The number operator $N$ ) Let $\tau\in(S\otimes S)^{*}$ be the trace operator
defined by$\eta\in S$ .
,$\kappa$$\mathrm{K}.\mathrm{R}$$\mathrm{P}.\mathrm{A}$$\langle\tau, \xi\otimes\eta\rangle=\langle\xi, \eta\rangle$$\xi,$The number operator $N$ is usually expressed as$\int_{\mathbb{R}}\partial_{t}*\partial_{t}dt$

by Kuo’s notation in white noise analysis. By the Obata theory, $N$ has the following
representation as a continuous linear operator from $(S)$ into itself, namely,$N= \Pi_{1,1}(\tau)=\int_{\mathbb{R}^{2}}\tau(S, t)\partial^{*}\partial tdsdtS^{\cdot}$

Example 10. (The Gross Laplacian
analysis we have the expression$\Delta_{G}$) By the usual notation in white noise$\Delta_{G}=\int_{\mathbb{R}}\partial_{t}^{2}dt$.

49Then the Gross Laplaciancan be also expressed by$\Delta_{G}$$\Delta_{G}=\Pi_{0,2}(_{\mathcal{T}})=\int_{\mathbb{R}^{2}}\tau(s_{1}, S2)\partial_{s}\partial_{s_{2}}dS1dS_{2}1$

as a continuous linear operator from $(S)$ into $(S)$ .
Let us consider the general expansion of our Pseudo-Fourier-Mehler transform.
We may take advantage of Obata’s integral kernel operator theory in order to obtainand its adjoint. That is to say,
Fock expansion representations of$G_{\theta}$$\Psi_{\theta}$

Theorem 21. For
the following Fock$\theta\in \mathbb{R}$, the $PFM$ transform:$\Psi_{\theta}$

and the adjoint operator$G_{\theta}$

have$expansi_{\mathit{0}}n\mathit{8}$(i)$\Psi_{\theta}=\sum_{=l,,m0}^{\infty}\frac{1}{l!m!}(ie\sin\theta)i\theta l(e-i\theta 1)^{m}\cdot\Pi 2l+m,m(\tau^{\otimes l}\otimes\lambda_{m})$(ii)$G_{\theta}= \sum_{=l,,m0}^{\infty}\frac{1}{l!m!}(ie\sin\theta i\theta)m(e-i\theta 1)^{l}\cdot \mathrm{I}\mathrm{I}_{l,l}+2m(\lambda l^{\otimes}\mathcal{T}\otimes m)$where the kernel$\lambda_{m}\in(S^{\otimes 2m})^{*}i_{\mathit{8}}$;;given by$\lambda_{m}.\cdot:=,.\sum_{m}^{\infty}e_{i_{1}}i_{1}.i_{2},\cdots,i=0\otimes\cdots\otimes e_{i_{m}}\otimes e_{i_{1}}\otimes\cdots\otimes e_{i_{m}}$.

\S 8. Generalization
Let $GL((S))$ be the group of all linear homeomorphisms from $(S)$ into $(S)$ . Then
we have

is a regular one para.meter sub.group of $GL((S))$ with
Proposition 22.
$infinite\mathit{8}imal$ generator $i(N+\Delta_{G})$ .
$\{G_{\theta;}\theta\in \mathbb{R}\}$Let us consider some generalization. Suggested by [1], for example we propose
whose
as operator on
to define the generalized PFM transform-functional is given by$X_{\theta},$$(S)^{*}$$\theta\in \mathbb{R}$$\mathrm{U}$

,

(32)

$S(X_{\theta} \Phi)(\xi)=\langle\langle\Phi, \exp(\mathrm{e}\langle\alpha\theta., \xi\rangle-\frac{1}{2}J(\alpha, \beta;\theta)|\xi|2)\rangle\rangle$(cf. Eq. (7) in Definition 1 of PFM transform), for $\xi\in S,$ $\Phi\in(S)^{*}$ . We set$J(\alpha, \beta;\theta)=\mathrm{e}^{2}-\alpha\theta 2H(\alpha, \beta;\theta)$

with

where
by
of

$h(\alpha, \beta)=\beta/2\alpha$$X_{\theta}$$Z_{\theta}$

, for

,

$H(\alpha, \beta;\theta)=h(\alpha, \beta)\cdot(\mathrm{e}^{2\alpha\theta}-1)$$\alpha,$$\beta\in \mathbb{C},$$\alpha\neq 0$,. Then we denote the adjoint operator.

Claim 23. The set$\{Z_{\theta;}\theta\in \mathbb{R}\}$is a regular one parameter subgroup of $GL((S))$ .

50

Claim 24. The infinitesimal generator of$+\beta\Delta_{G}$$\{z_{\theta;}\theta\in \mathbb{R}\}$.

Claim 25. The generalized $PFM$ transformgroup of $GL((S)^{*})$ .

Claim 26. The infinitesimal generator of$+\beta\Delta_{G}^{*}$

is given by the operator$\{X_{\theta;}\theta\in \mathbb{R}\}$$\{X_{\theta;}\theta\in \mathbb{R}\}$

is

$a$

$\alpha N$

one parameter sub-is given by the operator$\alpha N$.

Remark 11. The above definition (32) of generalized PFM transform
alternatively replaced by the following expression:$S(X_{\theta}\Phi)(\xi)=F(\mathrm{e}\xi\alpha\theta)\cdot\exp(H(\alpha, \beta;\theta)\cdot|\xi|^{2})$$X_{\theta}$

can be,, i.e., $S\Phi=F$ .
where $F$ denotes the -functional of in, then the above-defined generalized
Remark 12. Especially whenare, of course, attributed to the simple PFM transforms
PFM transforms
given by (6), (7) in the section 3.
$\mathrm{U}$$\Phi$$(S)^{*}$$\alpha=\beta=i(\in \mathbb{C})$$X_{\theta}$$\Psi_{\theta}$

Acknowledgements. The author wishes to express his sincere gratitude to Pro. Chung for their kind sugfessor I. Kubo, Professor N. Obata and Professor
gestions and many useful comments and discussions. The author is also gratefulfor the hospitality of RIMS, Kyoto University during March 29-31, 1995, International Institute for Advanced Studies (IIAS, Kyoto) during May 27-30, 1995,
SPA Conference) during June 19-23, 1995,
National University of Singapore (and Training Institute of Meiji MLIC (Japan-Russia Symposium) during July 2630, 1995. The results on PFM transform and its generalization under fully general
setting of white noise analysis will appear in our next paper.
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