数理解析研究所講究録 1227 巻 2001 年 184-198

184
Multi-Variable White Noise Functions:Standard Setup Revisited
NOBUAKI OBATAGRADUATE SCHOOL OF INFORMATION SCIENCESTOHOKU UNIVERSITYSENDAI 980-8579 JAPANobata@math.is.tohoku.ac.jp

1Introduction
In the recent development of white noise theory the framework proposed by CochranKuO-Sengupta [4] has become more important for their characterization theorems, seealso [1]. In fact, much attention has been paid to characterization theorems for the test
, for white noise operators
andfunctions , for the generalized functions. As was pointed out first by Chung-Chung-Ji [2], those characterization
fortheorems are related each other however the statements are not unified because theirobjects are different so far as we are concerned with asingle CKS-space over aparticularunderlying Gelfand triple. In this paper, using the standard setup of white noise calculus
proposed by Hida-Obata-Sait\^o [6] and by Obata [12], we unify those characterization
theorems into asingle statement.
Let $S_{A}(U)$ and $S_{B}(V)$ be countable Hilbert nuclear spaces constructed from $L^{2}(U)$and $L^{2}(V)$ in the standard manner, respectively, see 52. Then, for two weight sequences
we consider CKS-spaces of test white noise functions
anddefined by$\mathcal{W}^{*}$$\mathcal{W}$$\mathcal{L}(\mathcal{W}, \mathcal{W}^{*})$$\mathcal{L}(\mathcal{W}, \mathcal{W})$$\alpha=\{\alpha(n)\}_{n=0}^{\infty}$$\beta=\{\beta(n)\}_{n=0}^{\infty}$$\mathcal{U}\equiv\Gamma_{\alpha}(S_{A}(U))$,$\mathcal{V}\equiv\Gamma_{\beta}(S_{B}(V))$.

for
We assume condition (HI) for both $S_{A}(U)$ and $S_{B}(V)$ and conditions
in
the
is
statedresult
the
main
Then
in
\S \S 2-3.
and . These conditions are describedfollowing$(\mathrm{A}1)-(\mathrm{A}4)$$\alpha$$\beta$Theorem 1.1 For a continuous operator$(\xi, \eta)=\langle\langle_{-}^{-}-\phi_{\xi}, \phi_{\eta}\rangle\rangle$,

are exponential vectors in
and
there
following trno conditions:$\phi_{\eta}$$\phi_{\xi}$put$—\in \mathcal{L}(\mathcal{U}, \mathcal{V})$$\xi\in S_{A}(U)$$\mathcal{U}$,$\eta\in S_{B}(V)$,(1)

and , respectively. Then, $$ satisfies the$\mathcal{V}$(01) $$ is a G\^ateaux-entire function on $S_{A}(U)\cross S_{B}(V)$ ;(02) for any $p\geq 0$ there exist $q\geq 0$ and $C\geq 0$ such that$|(\xi, \eta)|^{2}\leq CG_{\alpha}(|\xi|_{p+q}^{2})G_{1/\beta}(|\eta|_{-p}^{2})$

where
and

$\xi\in S_{A}(U)$,$\eta\in S_{B}(V)$,are exponential generating functions of the weight sequences aand, respectively.

$G_{\alpha}$$1/\beta$,$G_{1/\beta}$

185
eConversely, if a -valued function defined on $8_{A}(U)\mathrm{x}5_{B}(V)$ fulfills conditions (01)and (02), there exists a unique continuous operator EC $C(U, 1^{7})$ satisfying (1).
$\mathrm{C}$Specializing the underlying spaces $S_{A}(U)$ and $S_{B}(V)$ , we obtain the characterization
theorems mentioned at the beginning as corollaries of our main theorem. Moreover,
our theorem yields characterization theorems for multi-variable white noise functions inahighly general form. It is noteworthy that multi-variable white noise functions havebecome more important in applications [13]. Further development is expected together
with another type of characterization theorems based on Bargmann-Segal spaces, which
has been also extensively studied along with complex white noise [7].

2Standard Construction of an Underlying Gelfand Triple
We assemble some notions and results in [12]. Let be atopological space equippedwith aa-finite Borel measure and consider the complex Hilbert space $L^{2}(U)=L^{2}(U, \nu)$ .
Let be aselfadjoint operator in $L^{2}(U)$ such that
. With each
we associate aHilbert space $E_{p}(U)$ with anorm defined by
$U$

$\nu$

$A$

$\inf \mathrm{S}\mathrm{p}\mathrm{e}\mathrm{c}(A)>0$$|\xi|_{p}=|A^{p}\xi|_{L^{2}(U)}$,$p\in \mathrm{R}$$p\in \mathrm{R}$.

More precisely, for $p\geq 0$ , $E_{p}(U)$ consists of 46 $L^{2}(U)$ satisfying $|\xi|_{p}<\infty$ and $E_{-p}(U)$. Thus we come to acountable
is the completion of $L^{2}(U)$ with respect to the norm
Hilbert space:$|\cdot$$S_{A}(U)=\mathrm{p}\mathrm{r}\mathrm{o}\mathrm{j}$$|_{-p}$$\lim E_{p}(U)$.

$parrow\infty$

The strong dual space of $S_{A}(U)$ is identical to the inductive limit:$S_{A}^{*}(U)= \mathrm{i}\mathrm{n}\mathrm{d}\lim_{arrow p\infty}E_{-p}(U)$and we come to arigging:$S_{A}(U)\subset L^{2}(U)\subset S_{A}^{*}(U)$(2).

The canonical -bilinear form on $S_{A}^{*}(U)\cross S_{A}(U)$ is denoted by.
the norm of $L^{2}(U)$ is given by$\mathrm{C}$$\langle\cdot$,$\cdot\rangle$. Then by definition$|\xi|_{0}^{2}=\langle\overline{\xi}, \xi\rangle$

Lemma 2.1 A countable Hilbert space $S_{A}(U)$ defined as above is nuclear if and only if
is of Hilbert-Schmidt type.
there exists r $>0$ such that
$A^{-r}$As for aspectral property of(01)$\inf \mathrm{S}\mathrm{p}\mathrm{e}\mathrm{c}(A)>1$

and$A^{-r}$$A$

we consider

is of Hilbert-Schmidt type for some $r>0$ .

The first condition is taken into account beforehand. It follows from (HI) that$||A^{-1}||_{\mathrm{o}\mathrm{P}}<1$,$\lim_{rarrow\infty}||A^{-r}||_{\mathrm{H}\mathrm{S}}=0$.

(3)

Definition Acountable Hilbert nuclear space $S_{A}(U)$ constructed from $L^{2}(U)$ by meansof aselfadjoint operator satisfying (HI) is called standard
$A$

186
As is suggested by (2), it is natural to consider $S_{A}(U)$ andare spaces of test
functions and generalized functions (or distributions) on , respectively. However, it is
not clear at all whether the delta function (evaluation map) is amember of. On
the contrary, continuity of atest function does not follow automatically. By construction
each element of $S_{A}(U)$ is merely afunction onwhich is determined up to -nullfunctions. We thus come to:$S_{A}^{*}(U)$$U$$S_{A}^{*}(U)$

$U$

$\nu$

(H2) for each function $\xi\in S_{A}(U)$ there exists aunique continuous functionthat $\xi(u)=\tilde{\xi}(u)$ for. $u\in U$ .

$\overline{\xi}$

on

$U$

such

$\nu- \mathrm{a}.\mathrm{e}$Once this condition is satisfied, we consider $S_{A}(U)$ always as aspace of continuousfunctions on and we do not use the exclusive symbol . Under (H2) we put two more
hypotheses to keep adelta function in:$U$$\tilde{\xi}$$S_{A}^{*}(U)$(H3) for each $u\in U$ the evaluation map
functional, i.e.,;$\delta_{u}$:$\xi-\rangle$$\xi(u)$, $\xi\in S_{A}(U)$ , is acontinuous linear$\delta_{u}\in S_{A}^{*}(U)$(H4) the map }topology of$u$$arrow\delta_{u}\in S_{A}^{*}(U)$$S_{A}^{*}(U)$.

, $u\in U$ , is continuous with respect to the strong dual

were also discussed by KubO-Takenaka [11]. These
Conditions similar tohypotheses are essential for topological arguments of $S_{A}(U)$ . Here we recall the following$(\mathrm{H}1)-(\mathrm{H}4)$

Proposition 2.2 Let $S_{A}(U)$ be a standard countable Hilbert space and let $\xi_{n}\in S_{A}(U)$ ,, be a sequence converging to 0in $S_{A}(U)$ . If (H2) and (H3) are satisfied, then$n=1,2$ ,the sequence converges pointwisely, $i.e.$ , $\lim_{narrow\infty}\xi_{n}(u)=\xi(u)$ for any $u\in U$ . Moreover,if (H4) is satisfied in addition, the poinrwise convergence is uniform on every compact
subset of .
$\cdots$$U$

Proposition 2.3 Let $S_{A}(U)$ be a standard countable Hilbert space satisfying conditions(H2) and (H3) Then,$||A^{-r}||_{\mathrm{H}\mathrm{S}}^{2}= \int_{U}|\delta_{u}|_{-r}^{2}\nu(du)=\sum_{j=0}^{\infty}\lambda_{j}^{-2r}<\infty$where$1<\lambda_{1}\leq\lambda_{2}\leq\cdots$,

are the eigenvalues of .
$A$

in aHilbert space $H_{i}(i=1,2)$ their
Recall that for two selfadjoint operatorsin acanonical way.
becomes aselfadjoint operator intensor productas well.
for $i=1,2$ , thenMoreover, if$A_{i}$$H_{1}\otimes H_{2}$$A_{1}\otimes A_{2}$$\inf \mathrm{S}\mathrm{p}\mathrm{e}\mathrm{c}(A_{1}\otimes A_{2})>0$$\inf \mathrm{S}\mathrm{p}\mathrm{e}\mathrm{c}(A_{i})>0$

Proposition 2.4 Let $S_{A}(U)$ and $S_{B}(V)$ be standard countable Hilbert nuclear spaces.
Then the canonical isomorphism $L^{2}(U\cross V)\cong L^{2}(U)\otimes L^{2}(V)$ induces a topologicalisomorphism:$S_{A\otimes B}(U\cross V)\cong S_{A}(U)\otimes S_{B}(V)$Moreover, if both $S_{A}(U)$ and $S_{B}(V)$ satisfy hypotheses.

$(H\mathit{2})-(H\mathit{4})$(4), so does$S_{A\otimes B}(U\cross V)$.

are known, see [12, Q1.4]. In fact, theseUseful sufficient conditions for, however, inconditions are essential to formulate quantum white noise
this paper we do not go into this subject$(\mathrm{H}2)-(\mathrm{H}4)$$\{a_{u}, a_{u}^{*} ; u\in U\}$

187
3Conditions for Weight Sequences
After
Asequence

we introduce some general notion for positive sequences.
of positive numbers is called $log$ -concave if$\mathrm{A}\mathrm{s}\mathrm{a}\mathrm{i}-\mathrm{K}\mathrm{u}\mathrm{b}\mathrm{o}-\mathrm{K}\mathrm{u}\mathrm{o}[1]$$\alpha=\{\alpha(n)\}_{n=0}^{\infty}$$\alpha(n)\alpha(n+2)\leq\alpha(n+1)^{2}$

Two positive sequences
exist positive constants

and$\alpha=\{\alpha(n)\}_{n=0}^{\infty}$$K_{1}$,$K_{2}$,$M_{1}$,$n=0,1,2$ ,(A1) $\alpha(0)=1$ and, $M_{2}>0$ such that

(A2)$\lim_{narrow\infty}\{\frac{\alpha(n)}{n!}\}^{1/n}=0$,$n=0,1,2$ ,for some$\sigma\geq 1$$\alpha$.

;;(A3) ais equivalent to apositive sequence $\gamma=\{\gamma(n)\}$ such that
(A1)$\ldots$

we consider the following conditions:$\alpha=\{\alpha(n)\}_{n=0}^{\infty}$$\inf_{n\geq 0}\sigma^{n}\alpha(n)>0$.

are called equivalent if there$\beta=\{\beta(n)\}_{n=0}^{\infty}$$K_{1}M_{1}^{n}\alpha(n)\leq\beta(n)\leq K_{2}M_{2}^{n}\alpha(n)$For apositive sequence$\ldots$$\{\frac{\gamma(n)}{n!}\}$

is log-concave;

is equivalent to another positive sequence $\gamma=\{\gamma(n)\}$ such that$\{\frac{1}{n!\gamma(n)}\}$

islog-concave.

For example,
with $0\leq\beta<1$ and the Bell numbers of order satisfy the above
conditions [4].
The exponential generating functions of and 1/0 are defined by$(n!)^{\beta}$$k$$\alpha$$G_{\alpha}(t)= \sum_{n=0}^{\infty}\frac{\alpha(n)}{n!}t^{n}$,$G_{1/\alpha}(t)= \sum_{n=0}^{\infty}\frac{1}{n!\alpha(n)}t^{n}$,respectively. Both are entire holomorphic functions by (A1) and (A2). We next define$\overline{G}_{\alpha}(t)\equiv\sum_{n=0}^{\infty}t^{n}\frac{n^{2n}}{n!\alpha(n)}\{\inf_{\tau>0}\frac{G_{\alpha}(\tau)}{\tau^{n}}\}$,$\overline{G}_{1/\alpha}(t)\equiv\sum_{n=0}^{\infty}t^{n}\frac{n^{2n}\alpha(n)}{n!}\{\inf_{\tau>0}\frac{G_{1/\alpha}(\tau)}{\tau^{n}}\}$.

(A3) and (A4) are necessary and sufficient conditions respectively
It is known [1]to have positive radiai of convergence. These functions will play aforand forinnorm estimates. Moreover the next fact is known [1].
crucial role$\underline{\mathrm{t}}\mathrm{h}\mathrm{a}\mathrm{t}$$\overline{G}_{\alpha}$$G_{1/\alpha}$

Lemma 3.1 Assume that $\alpha=\{\alpha(n)\}$ satisfies $(Al)-(A\mathit{4})$ .
(1) There exists a constant$C_{\alpha 1}>0$such that$\alpha(n)\alpha(m)\leq C_{\alpha 1}^{n+m}\alpha(n+m)$(2) There exists a constant$C_{\alpha 2}>0$,$n$, $m=0,1,2$ ,,$n$, $m=0,1,2$ ,$\ldots$.

such that$\alpha(n+m)\leq C_{\alpha 2}^{n+m}\alpha(n)\alpha(m)$$\ldots$.

188(3) There exists a constant$C_{\alpha 3}>0$

such that$\alpha(n)\leq C_{\alpha 3}^{m}\alpha(m)$,$0\leq n\leq m$.

Then, by asimple calculation we have

Proposition 3.2 Let $\alpha=\{\alpha(n)\}$ be as above andfunction. Then, for , $t\geq 0$ we have:$G_{\alpha}(t)$the exponential generating$s$(1) Ga (t) $=1$ and Ga(s)for$\leq G_{\alpha}(t)$$G_{\alpha}(s)G_{\alpha}(t)\leq G_{\alpha}(C_{\alpha 1}(s+t))$(3)$G_{\alpha}(s+t)\leq G_{\alpha}(C_{\alpha 2}s)G_{\alpha}(C_{\alpha 2}t)$$(\mathrm{s}+t))$

.

.

(2)

(4) esGa{t)\leq Ga{Ca3$s\leq t$.

, in particular,$e^{t}\leq G_{\alpha}(C_{\alpha 3}t)$.

4Standard CKS-Space
Suppose we are given astandard, countable Hilbert nuclear space:$S_{A}(U)=\mathrm{p}\mathrm{r}\mathrm{o}\mathrm{j}$$\lim E_{p}(U)$$parrow\infty$

and apositive sequence $\alpha=\{\alpha(n)\}$ satisfying
Fock space:$\Gamma_{\alpha}(E_{p}(U))=\{\phi=(f_{n})_{n=0}^{\infty}$

where
limit:$E_{p}(U)^{\otimes n}\wedge$;$f_{n}\in E_{p}(U)^{\otimes n}\wedge$$(\mathrm{A}1)-(\mathrm{A}4)$,. We first form aweighted$|| \phi||_{p}^{2}\equiv\sum_{n=0}^{\infty}n!\alpha(n)|f_{n}|_{p}^{2}<\infty\}$,stands for the -fold symmetric tensor power, and then take its projective$n$$\mathcal{U}=\Gamma_{\alpha}(S_{A}(U))=\mathrm{p}\mathrm{r}\mathrm{o}\mathrm{j}\lim\Gamma_{\alpha}(E_{p}(U))$.

$parrow\infty$Since$\mathcal{U}$becomes acountable Hilbert nuclear space, we obtain aGelfand triple:$\mathcal{U}=\Gamma_{\alpha}(S_{A}(U))\subset\Gamma(L^{2}(U))\subset\Gamma_{\alpha}(S_{A}(U))^{*}=\mathcal{U}^{*}$(5),

where the middle space is the usual Boson Fock space, i.e., aweighted Fock space with. We may consider (5) as avariant of “second quantization”
weight sequenceof an underlying Gelfand triple:$\alpha(n)\equiv 1$$S_{A}(U)\subset L^{2}(U)\subset S_{A}^{*}(U)$.

We call (5) astandard $CKS$-space after [4].
The topology of is defined by the family of norms:$\mathcal{U}$$|| \phi||_{p,+}^{2}=\sum_{n=0}^{\infty}n!\alpha(n)|f_{n}|_{p}^{2}$,$\=(f_{n})$c&,$p\geq 0$.

189
The canonical$\mathrm{C}$-bilinear form on$\mathcal{U}^{*}\cross \mathcal{U}$$\langle\langle\Phi, \phi\rangle\rangle=\sum_{n=0}^{\infty}n!\langle F_{n}, f_{n}\rangle$,

is denoted by$\Phi$$\langle\langle\cdot, \cdot\rangle\rangle$$=(F_{n})\in \mathcal{U}^{*}$,. Then$\phi$$=(f_{n})\in \mathcal{U}$,and it holds that$|\langle\langle\Phi, \phi\rangle\rangle|\leq||\Phi||_{-p,-}||\phi||_{p,+}$

,

where
$|| \Phi||_{-p,-}^{2}=\sum_{n=0}^{\infty}\frac{n!}{\alpha(n)}|F_{n}|_{-p}^{2}$,$\Phi$$=(F_{n})\in \mathcal{U}^{*}$.

We also note that$\Gamma_{\alpha}(S_{A}(U))^{*}\cong \mathrm{i}\mathrm{n}\mathrm{d}\lim_{arrow p\infty}$$\Gamma_{1/\alpha}(E_{-p}(U))$

,

where
carries the strong dual topology andfor atopological linear
isomorphism.
Conditions
are sorted out by
from many similar ones
that have been introduced to keep “nice” properties of aCKS-space. On the other hand,
it is also possible to start with agenerating function
or another function controlling
growth rate. This reversed approach is concise and useful for some questions [1], [5], [8],however, we prefer to the explicit description for our later calculation.
By the Wiener-It\^o-Segal isomorphism the Boson Fock space $\Gamma(L^{2}(U))$ is realized
as an -space over aGaussian space. In that sense
is aspace of functions
on the Gaussian space. By aparallel argument with [12, \S 3.2] one can show easily. Thus, for example, white noise delta
thatsatisfies conditions
functions are defined as white noise distributions.
$\Gamma_{\alpha}(S_{A}(U))^{*}$$\cong \mathrm{s}\mathrm{t}\mathrm{a}\mathrm{n}\mathrm{d}\mathrm{s}$$(\mathrm{A}1)-(\mathrm{A}4)$$\mathrm{A}\mathrm{s}\mathrm{a}\mathrm{i}-\mathrm{K}\mathrm{u}\mathrm{b}\mathrm{o}-\mathrm{K}\mathrm{u}\mathrm{o}[1]$$G_{\alpha}$$L^{2}$$\Gamma_{\alpha}(S_{A}(U))$$\Gamma_{\alpha}(S_{A}(U))$$(\mathrm{H}2)-(\mathrm{H}4)$

5Proof of Main Theorem
First we recall some notation used in the statement of Theorem 1.1. For each
$S_{A}(U)$ we put$\phi_{\xi}=(1,$$\frac{\xi}{1!}$,$\frac{\xi^{\otimes 2}}{2!}$,$\ldots$,$\frac{\xi^{\otimes n}}{n!}$,$\ldots$$\xi\in$).

,beand is called an exponential vector or acoherent vector. Let ,Then$F$ defined onfunctionThena-valuedingeneral.
spacescomplex topological vector
is called G\^ateaux-entire if the function$x_{1}$$\phi_{\xi}\in \mathcal{U}$$\ldots$$x_{n}$$\mathrm{C}$$x_{1}\cross\cdots\cross x_{n}$$z\vdash+F(\xi_{1}, \ldots, \xi_{k}+z\xi’, \ldots, \xi_{n})$is entire holomorphic in$z\in \mathrm{C}$for any choice of$1\leq k\leq n$ .

Now we start with the followin$\xi_{1}\in x_{1}$,$\ldots$,$\xi_{n}\in x_{n}$and $\xi’\in x_{k}$ ,190Lemma 5.1 Let $F:S_{A}(U)arrow \mathrm{C}$ be a G\^ateaux-entire function. Assume that there exist
such thatan entire function on and$G$$p\in \mathrm{R}$$\mathrm{C}$$|F(\xi)|^{2}\leq G(|\xi|_{p}^{2})$,$\xi\in S_{A}(U)$.

Then for any $n\geq 0$ the Gateaux derivative$F_{n}( \xi_{1}, \ldots, \xi_{n})=\frac{1}{n!}D_{\xi_{1}}\ldots$(6)$D_{\xi_{n}}F(0)$becomes a continuous -linear form on $S_{A}(U)$ satisfying$n$$|F_{n}|_{-(p+s)}^{2} \leq(\frac{n^{n}}{n!})^{2}\{\inf_{\tau>0}\frac{G(\tau)}{\tau^{n}}\}||A^{-s}||_{\mathrm{H}\mathrm{S}}^{2n}$.

(7)

is a(not necessarily continuous) n-linear. It is astandard result that
PROOF.
form on $S_{A}(U)$ .
. The Taylor expansion of an entire function }$arrow F(z\xi)$ is given by$F_{n}$$1^{\mathrm{o}}$$z$$2^{\mathrm{o}}$$F(z \xi)=\sum_{n=0}^{\infty}F_{n}(\xi, \ldots, \xi)z^{n}$$3^{\mathrm{o}}$,$\xi\in S_{A}(U)$.

. By Cauchy’s integral formula we obtain$|F_{n}( \xi, \ldots, \xi)|\leq|\xi|_{p}^{n}\{\inf_{\tau>0}\frac{G(\tau)}{\tau^{n}}\}^{1/2}$$4^{\mathrm{o}}$. By the polarization formula we have$\sup\{|F_{n}(\xi_{1}, \ldots, \xi_{n})|;|\xi_{1}|_{p}\leq 1, \ldots, |\xi_{n}|_{p}\leq 1\}\leq\frac{n^{n}}{n!}\{\inf_{\tau>0}\frac{G(\tau)}{\tau^{n}}\}^{1/2}$

the corresponding
be the eigenvalues of and. Let$L^{2}(U)$. Theneigenvectors which form acomplete orthonormal basis of$5^{\mathrm{o}}$$A$$1<\lambda_{0}\leq\lambda_{1}\leq\ldots$$|F_{n}|_{-(p+s)}^{2}= \sum_{j_{1},\ldots,j_{n}=0}^{\infty}|F_{n}(e_{j_{1}}$,...’$\{e_{j}\}_{j=0}^{\infty}$$e_{j_{n}})|^{2}\lambda_{j_{1}}^{-2(p+s)}\ldots\lambda_{j_{n}}^{-2(p+s)}$$= \sum_{j_{1},\ldots,j_{n}=0}^{\infty}|F_{n}(\lambda_{j_{1}}^{-p}e_{j_{1}}, \ldots, \lambda_{j_{n}}^{-p}e_{j_{n}})|^{2}\lambda_{j_{1}}^{-2s}\ldots\lambda_{j_{n}}^{-2s}$$\leq(\frac{n^{n}}{n!})^{2}\{\inf_{\tau>0}\frac{G(\tau)}{\tau^{n}}\}||A^{-s}||_{\mathrm{H}\mathrm{S}}^{2n}$.

1This completes the proof.

Lemma 5.2 Let $F:S_{A}(U)arrow \mathrm{C}$ be a G\^ateaux-entire function. Assume that there exist
such that
constants $C\geq 0$ and$p\in \mathrm{R}$$|F(\xi)|^{2}\leq CG_{\alpha}(|\xi|_{p}^{2})$theFor each $n\geq 0$ denote by
and we have$F_{n}$$n$,$\xi\in S_{A}(U)$.

-th G\^ateaux derivative defined by (6). Then (I)$(\mathrm{F}\mathrm{n})\in\Gamma_{\alpha}(S_{A}(U))^{*}$$||\Phi||_{-(p+s),-}^{2}\leq C\tilde{G}_{\alpha}(||A^{-s}||_{\mathrm{H}\mathrm{S}}^{2})$

.

$=$

191
Proof.

By the definition of norms and (7) we have$|| \Phi||_{-(p+s),-}^{2}=\sum_{n=0}^{\infty}\frac{n!}{\alpha(n)}|F_{n}|_{-(p+s)}^{2}$$\leq\sum_{n=0}^{\infty}\frac{n!}{\alpha(n)}(\frac{n^{n}}{n!})^{2}\{\inf_{\tau>0}\frac{CG_{\alpha}(\tau)}{\tau^{n}}\}||A^{-s}||_{\mathrm{H}\mathrm{S}}^{2n}$$=C\tilde{G}_{\alpha}(||A^{-s}||_{\mathrm{H}\mathrm{S}}^{2})$,as desired.

$\mathrm{I}$Recall thathas apositive radius of convergence. Thenfor
all sufficiently large $s>0$ becauseby (3). In asimilar manner wehave$\overline{G}_{\alpha}$$\overline{G}_{\alpha}(||A^{-s}||_{\mathrm{H}\mathrm{S}}^{2})<\infty$$\lim_{sarrow\infty}||A^{-s}||_{\mathrm{H}\mathrm{S}}=0$

Lemma 5.3 Let $F:S_{A}(U)arrow \mathrm{C}$ be a G\^ateaux-entire function. Assume that there exist
constants $C\geq 0$ andsuch that$p\in \mathrm{R}$$|F(\xi)|^{2}\leq CG_{1/\alpha}(|\xi|_{p}^{2})$For each $n\geq 0$ let
be theand we have$F_{n}$$n$,$\xi\in S_{A}(U)$.

-th G\^ateaux derivative defined by (6). Then $\Phi=(F_{n})\in$$\Gamma_{\alpha}(S_{A}(U))^{*}$.

$||\Phi||_{-(p+s),+}^{2}\leq C\overline{G}_{1/\alpha}(||A^{-s}||_{\mathrm{H}\mathrm{S}}^{2})$Remark Lemma 5.2 is the s0-called characterization theorem for white noise distributions first shown by Potthoff-Streit [15] for aparticular CKS-space called the HidaKubO-Takenaka space. The essence of their proof, however, remains invariable thoughafew variants have been discussed during the recent development of white noise theory.
Lemma 5.3 implies immediately characterization theorem for white noise test functions. In this connection see also Theorem 6.1.
$\Gamma_{\alpha}(S_{A}(U))$

Proof OF THEOREM 1.1. Fix $\eta\in S_{B}(V)$ and we consider$F_{\eta}(\xi)=(\xi, \eta)$

Then by Lemma 5.2 there exists$\Phi_{\eta}\in \mathcal{U}^{*}$$F_{\eta}(\xi)=\langle\langle\Phi_{\eta}, \phi_{\xi}\rangle\rangle$,,$\xi\in S_{A}(U)$.

such thati.e.,,$(\xi, \eta)=\langle\langle\Phi_{\eta}, \phi_{\xi}\rangle\rangle$

and$||\Phi_{\eta}||_{-(p+q+s),-}^{2}\leq C\overline{G}_{\alpha}(||A^{-s}||_{\mathrm{H}\mathrm{S}}^{2})G_{1/\beta}(|\eta|_{-p}^{2})$Next, for afixed$\phi\in \mathcal{U}$.

we consider$H_{\phi}(\eta)=\langle\langle\Phi_{\eta}, \phi\rangle\rangle$,$\eta\in S_{B}(V)$.

Obviously,$|H_{\phi}(\eta)|^{2}\leq||\Phi_{\eta}||_{-(p+q+s),-}^{2}||\phi||_{p+q+s,+}^{2}$$\leq C\overline{G}_{\alpha}(||A^{-s}||_{\mathrm{H}\mathrm{S}}^{2})G_{1/\beta}(|\eta|_{-p}^{2})||\phi$$||_{p+q+s,+}^{2}$.

192
Moreover,is acompact uniform limit of asequence of entire functions which
are linear combinations of$H_{\ovalbox{\tt\small REJECT}\ovalbox{\tt\small REJECT}\rangle}(\mathrm{t}7+\mathrm{z}\mathrm{y}7’)$$H_{\phi_{\xi}}(\eta+z\eta’)=\langle\langle\Phi_{\eta+z\eta’}, \phi_{\xi}\rangle\rangle=(\xi, \eta+z\eta’)$where 4runs over $S_{A}(U)$ , and hence
we find Qp 6
such that$H_{\phi}$,

is G\^ateaux-entire. Applying Lemma 5.3 to$H_{\phi}$$\mathcal{V}^{*}$,$H_{\phi}(\eta)=\langle\langle\Psi_{\phi}, \phi_{\eta}\rangle\rangle$$\eta\in S_{B}(V)$and$||\Psi_{\phi}||_{-(-p+t),+}^{2}\leq C\tilde{G}_{\alpha}(||A^{-s}||_{\mathrm{H}\mathrm{S}}^{2})\tilde{G}_{1/\beta}(||A^{-t}||_{\mathrm{H}\mathrm{S}}^{2})||\phi||_{p\dagger q+s,+}^{2}$

Define alinear map –:-$\mathcal{U}arrow \mathcal{V}$by$—\phi=\Psi_{\phi}$$\mathrm{t}\mathrm{h}\mathrm{a}\mathrm{t}---\mathrm{i}\mathrm{s}$$—\in \mathcal{L}(\mathcal{U}, \mathcal{V})$,continuous. Since$\langle\langle_{-}^{-}-\phi_{\xi}, \phi_{\eta}\rangle\rangle=H_{\phi_{\xi}}(\eta)=\langle\langle\Phi_{\eta}, \phi_{\xi}\rangle\rangle=(\xi, \eta)$

this

(8)

. Then (8) implies that$||---\phi||_{p-t,+}^{2}\leq C\tilde{G}_{\alpha}(||A^{-s}||_{\mathrm{H}\mathrm{S}}^{2})\overline{G}_{1/\beta}(||A^{-t}||_{\mathrm{H}\mathrm{S}}^{2})||\phi||_{p+q+s,+}^{2}$

which proves.

,is what we searched for.

I

In fact, as the above proof is asimple combination of [12, \S 3.6] and [2], nothing
new is required essentially. Afew variants of the proof are easily obtained following thearguments [1], [3], [4], [5], [9], [10].

6Unification of Traditional Characterization Theorems
are linearly independent and span a
Since the exponential vectors
dense subspace of , they play afundamental role in specifying white noise functions
and white noise operators. In practice, the most important are the -transform ofdefined by$\{\phi_{\xi} ; \xi\in E_{\mathrm{c}}\}$$\mathcal{U}$$S$$\Phi\in \mathcal{U}^{*}$$S\Phi(\xi)=\langle\langle\Phi, \phi_{\xi}\rangle\rangle$

and the symbol of$—\in \mathcal{L}(\mathcal{U},\mathcal{U}^{*})$,$\xi\in S_{A}(U)$,defined by$–(-\wedge\xi, \eta)=\langle\langle_{-}^{-}-\phi_{\xi}, \phi_{\eta}\rangle\rangle$

,

$\xi$

, $\eta\in S_{A}(U)$ .

The traditional characterization theorems are mentioned as follows.
Theorem 6.1 A -valued function F defined on $S_{A}(U)$ is theif and only if$\mathrm{C}$$\Phi\in \mathcal{U}^{*}$(F1) $F$ is G\^ateaux-entirei(F2) there exist $C\geq 0$ and $p\geq 0$ such that$|F(\xi)|^{2}\leq CG_{\alpha}(|\xi|_{p}^{2})$,$\xi\in E_{\mathrm{C}}$.

$S$-transform of some

193
Moreover,$\Phi$$\in \mathcal{U}$if and only if (Fl) and(F3) for any $p\geq 0$ there exist $C\geq 0$ such that$|F(\xi)|^{2}\leq CG_{1/\alpha}(|\xi|_{-p}^{2})$Theorem 6.2 A
operator$\mathrm{C}$,$\xi\in E_{\mathrm{C}}$.

-valued function 0defined on $S_{A}(U)\cross S_{A}(U)$ is the symbol of an
if and only if$—\in \mathcal{L}(\mathcal{U}, \mathcal{U}^{*})$(01) $$ is G\^ateaux-entire;(02) there exist constant numbers $C\geq 0$ and $p\geq 0$ such that$|(\xi, \eta)|^{2}\leq CG_{\alpha}(|\xi|_{p}^{2})G_{\alpha}(|\eta|_{p}^{2})$Moreover,$\cup--\in \mathcal{L}(\mathcal{U},\mathcal{U})$

,

$\xi$

, $\eta\in S_{A}(U)$ .

if and only if (01) and(F3) for any $p\geq 0$ there exist constant numbers $C\geq 0$ and $q\geq 0$ such that$|(\xi, \eta)|^{2}\leq CG_{\alpha}(|\xi|_{p+q}^{2})G_{1/\alpha}(|\eta|_{-p}^{2})$

,

$\xi$

, $\eta\in S_{A}(U)$ .

We may regard the zero space {0} also as astandard countable Hilbert space though
there is no underlying topological space or rather we may understand that $S(\emptyset)=\{0\}$ .
Then
is one-dimensional space consisting only scalar multiples of the vacuum
vector . As for Theorem 6.1, if we set $S_{B}(V)=\{0\}$ in Theorem 1.1, the statement
is characterization of $S_{A}(U)^{*}$ . If we set $S_{A}(U)=\{0\})$ the statement is characterization. The second part of Theorem 6.2 is immediate from Theorem 1.1 by settingof$S_{A}(U)=S_{B}(V)$ . We need some discussion to derive the first part of Theorem 6.2.
$U$$\Gamma_{\alpha}(S(\emptyset))$$\phi_{0}$$S_{B}(\ddagger^{f})$Lemma 6.3 If $S_{A}(U)$ and $S_{B}(V)$ are standard countable Hilbert nuclear spaces, so is
and $S_{A\oplus B}(U\cup V)\cong S_{A}(U)\oplus S_{B}(V)$ .

$S_{A\oplus B}(U\cup V)$

Lemma 6.4 If $S_{A}(U)$ is a standard countable Hilbert nuclear space, so isfor any finite discrete space equipped with counting measure. Moreover,, where $|T|=n$ .
$T$$S_{A\otimes I}(U\cross T)$$S_{A\otimes I}(U\cross T)\cong$$S_{A}(U)\otimes \mathrm{C}^{n}$

The above results are straightforward. In general, for two Hilbert spaces
there is acanonical unitary isomorphism$H_{1}$

and$H_{2}$$\mathcal{T}:\Gamma(H_{1})\otimes\Gamma(H_{2})arrow\Gamma(H_{1}\oplus H_{2})$specified by the correspondence of exponential vectors$\phi_{\xi_{1}}$,$\mathcal{T}(\phi_{\xi_{1}},.

$)\xi_{J}=(0, \ldots, 0, \xi_{1}\otimes..\otimes\xi_{j}, 0, \ldots)\wedge.\wedge$,\xi_{j}\otimes\psi_{\eta_{1},\ldots,\eta k})$,$\mathcal{T}(\phi_{\xi}\otimes\phi_{\eta})=\phi_{\xi\oplus\eta}$. In fact, for$\psi_{\eta_{1},..,\eta k}=(0, \ldots, 0, \eta_{1}\otimes..\otimes\eta_{k}, 0, \ldots)\wedge.\wedge$

is given as$\mathcal{T}(\phi_{\xi_{1},\ldots,\xi_{j}}\otimes\psi_{\eta_{1},\ldots,\eta k})=(0, \ldots, 0, h_{j+k}, 0, \ldots)$

,

where
.

$h_{j+k}=(\xi_{1}\oplus 0)\otimes..\otimes(\xi_{j}\oplus 0)\otimes(0\oplus\eta_{1})\otimes\ldots\otimes(0\oplus\eta_{k})\wedge.\wedge\wedge\wedge\wedge$,

194
0, , 0, \ldots ) (:(0,
For arbitrary $$T(Cl)$ (&e) is given by abilinear map
that$\mathrm{I}^{\ovalbox{\tt\small REJECT}}(H_{1})$$\ovalbox{\tt\small REJECT} 7_{\mathrm{y}}|$$\ovalbox{\tt\small REJECT}$\ldots ,and$\ovalbox{\tt\small REJECT} I\ovalbox{\tt\small REJECT}$$h_{\ovalbox{\tt\small REJECT}^{\ovalbox{\tt\small REJECT}\ovalbox{\tt\small REJECT}}},H\mathit{7}^{j}\mathrm{x}H\mathit{7}^{k}-\ovalbox{\tt\small REJECT}=(H_{1}\mathrm{e} H_{2})^{\ovalbox{\tt\small REJECT}(7+k)}\ovalbox{\tt\small REJECT}$$\mathcal{T}(\phi\otimes\psi)=(0, \ldots, 0, h_{j,k}(f_{j}, g_{k}), 0, \ldots)$

For this$h_{j,k}$)’E ’in such away$(H_{2})\rangle$$(0_{\mathrm{j}}\ovalbox{\tt\small REJECT}\ovalbox{\tt\small REJECT} \ovalbox{\tt\small REJECT}\rangle\rangle 0g_{k\ovalbox{\tt\small REJECT}}0_{\ovalbox{\tt\small REJECT}}\ovalbox{\tt\small REJECT}\ovalbox{\tt\small REJECT}\ovalbox{\tt\small REJECT}$.

(9)

we have by Fourier expansion$|h_{j,k}(f_{j}, g_{k})|_{(H_{1}\oplus H_{2})^{\otimes(j+k)}}^{2}= \frac{j!k!}{(j+k)!}|f_{j}|_{H_{1}^{\otimes j}}^{2}|g_{k}|_{H_{2}^{\otimes k}}^{2}$

.

(10)

Lemma 6.5 The canonical isomorphism $\Gamma(L^{2}(U)\oplus L^{2}(V))\cong\Gamma(L^{2}(U))\otimes\Gamma(L^{2}(V))$
induces a topological isomorphism:$\Gamma_{\alpha}(S_{A}(U)\oplus S_{B}(V))\cong\Gamma_{\alpha}(S_{A}(U))\otimes\Gamma_{\alpha}(S_{B}(V))$.

be the canonical isomorphism from $\Gamma(L^{2}(U))\otimes\Gamma(L^{2}(V))$ onto
described above. For $\phi=(f_{j})\in\Gamma(L^{2}(U))$ and $=(g_{k})\in\Gamma(L^{2}(V))$ ,we set $\mathcal{T}(\phi\otimes\psi)=(h_{n})\in\Gamma(L^{2}(U)\oplus L^{2}(V))$ . Then by (9) we have
Proof.

Let$\mathcal{T}$$\Gamma(L^{2}(U)\oplus L^{2}(V))$$\psi$$h_{n}= \sum_{j+k=n}h_{j,k}(f_{j}, g_{k})$.

Since the right hand side is an orthogonal sum, using (10) we come to$|h_{n}|_{p}^{2}= \sum_{j+k=n}|h_{j,k}(f_{j}, g_{k})|_{p}^{2}=\sum_{j+k=n}\frac{j!k!}{(j+k)!}|f_{j}|_{p}^{2}|g_{k}|_{p}^{2}$

.

Hence
$|| \mathcal{T}(\phi\otimes\psi)||_{p,+}^{2}=\sum_{n=0}^{\infty}n!\alpha(n)|h_{n}|_{p}^{2}=\sum_{j,k=0}^{\infty}j!k!\alpha(j+k)|f_{j}|_{p}^{2}|g_{k}|_{p}^{2}$

.

(11)

Then by Lemma 3.1 (2),$|| \mathcal{T}(\phi\otimes\psi)||_{p,+}^{2}\leq\sum_{j=0}^{\infty}j!\alpha(j)C_{\alpha 2}^{j}|f_{j}|_{p}^{2}\sum_{k=0}^{\infty}k!\alpha(k)C_{\alpha 2}^{k}|g_{k}|_{p}^{2}$

Choose $q\geq 0$ such that

$C_{\alpha 2}||A^{-1}||_{\mathrm{o}\mathrm{P}}^{2q}\leq 1$

and$C_{\alpha 2}||B^{-1}||_{\mathrm{o}\mathrm{P}}^{2q}\leq 1$

.

(12)

. Then (12) becomes$|| \mathcal{T}(\phi\otimes\psi)||_{p,+}^{2}\leq\sum_{j=0}^{\infty}j!\alpha(j)|f_{j}|_{p+q}^{2}\sum_{k=0}^{\infty}k!\alpha(k)|g_{k}|_{p+q}^{2}=||\phi||_{p+q,+}^{2}||\psi||_{p+q,+}^{2}$,that is,$||\mathcal{T}(\phi\otimes\psi)||_{p,+}\leq||\phi||_{p+q,+}||\psi||_{p+q,+}$

.

(13)

In asimilar manner, applying Lemma 3.1 (1) to (11), we obtain$||\mathcal{T}(\phi\otimes\psi)||_{p,+}\geq||\phi||_{p-r,+}||\psi$ $||_{p-r,+}$

(14)

,

and
where $r\geq 0$ is taken in such away that
assertion then follows from (13) and (14).
there is no isomorphism between
andRemark For Hilbert spaces
for ageneral .
and$C_{\alpha 1}||A^{-1}||_{\mathrm{o}\mathrm{P}}^{2r}\leq 1$$H_{2}$$H_{1}$$\Gamma_{\alpha}(H_{1})\otimes\Gamma_{\alpha}(H_{2})$$\alpha$$C_{\alpha 1}||B^{-1}||_{\mathrm{o}\mathrm{P}}^{2r}\leq 1$

. The

1

$\Gamma_{\alpha}(H_{1}\oplus H_{2})$195
Lemma 6.6 Let $S_{A}(U)$ be a standard countable Hilbert nuclear space and let
discrete space of points with counting measure. Then the isomorphism

$T$

be

$a$

$n$

$\Gamma(L^{2}(U\cross T))\cong\Gamma(L^{2}(U))\otimes\cdots$(&Y $(L^{2}(U))$( times)$n$induces a topological isomorphism( times).

$\Gamma_{\alpha}(S_{A\otimes I}(U\cross T))\cong\Gamma_{\alpha}(S_{A}(U))\otimes\cdots\otimes\Gamma(S_{A}(U))$

PROOF.

$n$Note first that$L^{2}(T)=S_{I}(\dot{T})\cong \mathrm{C}^{n}$.

It then follows from Proposition 2.4 that the canonical isomorphism( times)$L^{2}(U\cross T)\cong L^{2}(U)\otimes L^{2}(T)\cong L^{2}(U)\oplus\cdots\oplus L^{2}(U)$

$n$

(15)

induces atopological isomorphism:( times).

$S_{A\otimes I}(U\cross T)\cong S_{A}(U)\otimes S_{I}(T)\cong S_{A}(U)\oplus\cdots\oplus S_{A}(U)$$n$On the other hand, from (15) we see that
( times)$\Gamma(L^{2}(U\cross T))\cong\Gamma(L^{2}(U))\otimes\cdots\otimes\Gamma(L^{2}(U))$$n$and from Lemma 6.5$\Gamma_{\alpha}(S_{A\otimes I}(U\cross T))\cong\Gamma_{\alpha}(S_{A}(U))$

C&$\cdots$$\otimes\Gamma(S_{A}(U))$( times).
$n$

I

The assertion is then clear.

We go back to Theorem 1.1. By Lemma 6.6 the statement of Theorem 1.1 remains
with 0. Moreover,valid if is replaced with $U\cross\{1,2\}$ and
$V$

$U$

$\mathcal{L}(\Gamma_{\alpha}(S_{A\otimes I}(U\cross\{1,2\})), \mathrm{C})\cong \mathcal{L}(\Gamma_{\alpha}(S_{A}(U))\otimes\Gamma_{\alpha}(S_{A}(U)), \mathrm{C})$$\cong\{\Gamma_{\alpha}(S_{A}(U))\otimes\Gamma_{\alpha}(S_{A}(U))\}^{*}$$\cong \mathcal{L}(\Gamma_{\alpha}(S_{A}(U)), \Gamma_{\alpha}(S_{A}(U))^{*})$.

Thus, in this case, Theorem 1.1 is reduced to characterizaion of white noise operators.
In order to complete the reduction we need to discuss conditions (01) and (02), see
Theorem 6.2. Let $T=\{1,2, \ldots, n\}$ . By the isomorphism
described in Lemma 6.4 we come to$\mathcal{L}(\Gamma_{\alpha}(S_{A}(U)), \Gamma_{\alpha}(S_{A}(U))^{*})$$S_{A\otimes I}(U\cross T)\cong S_{A}(U)\otimes \mathrm{C}^{n}$$S_{A\otimes I}(U\cross T)\cong S_{A}(U)\oplus\cdots\oplus S_{A}(U)$,For
given by$(\xi_{1}$$\ldots$:$\xi_{n})\in S_{A}(U)\oplus\cdots\oplus S_{A}(U)$( times).
$n$the corresponding element $\xi\in S_{A\otimes I}(U\cross T)$ is$\xi(u, j)=\xi_{j}(u)$Then there is aone-t0-0ne correspondence between functions on$S_{A}(U)\cross\cdots\cross S_{A}(U)$ (times) given by$n$$(\xi)=F(\xi_{1}, \ldots, \xi_{n})$.

$S_{A\otimes I}(U\cross T)$

and on196
eLemma 6.7 Notations being as above,
is
x
F
(n
times).
soisonif$5_{\ovalbox{\tt\small REJECT}^{\ovalbox{\tt\small REJECT}}\mathrm{t}}(U)$Proof.

$\ovalbox{\tt\small REJECT}\ovalbox{\tt\small REJECT}\ovalbox{\tt\small REJECT}$-entire on$Gc\ovalbox{\tt\small REJECT} teaur\ovalbox{\tt\small REJECT}$$\mathit{8}_{A(g)}(U$xT) if and only$\mathrm{x}\mathrm{J}_{A}(U)$I

We need only to recall Hartogs’ theorem of holomorphy.

Lemma 6.8 Notations being as above, if there exist $C\geq 0$ and$|(\xi)|^{2}\leq CG_{\alpha}(|\xi|_{p}^{2})$,$p\in \mathrm{R}$$\xi\in S_{A\otimes I}(U\cross T)$

such that,then there exists $q\geq 0$ such that$|F( \xi_{1}, \ldots, \xi_{n})|^{2}\leq C\prod_{j=1}^{n}G_{\alpha}(|\xi_{j}|_{p+q}^{2})$Conversely, if there exist $C\geq 0$ and$p\in \mathrm{R}$,$\xi_{1}$,$\ldots$, $\xi_{n}\in S_{A}(U)$ .

such that,$|F( \xi_{1}, \ldots, \xi_{n})|^{2}\leq C\prod_{j=1}^{n}G_{\alpha}(|\xi_{j}|_{p}^{2})$$\xi_{1}$,$\ldots$, $\xi_{n}\in S_{A}(U)$ ,then there exists $q\geq 0$ such that$|(\xi)|^{2}\leq CG_{\alpha}(|\xi|_{p+q}^{2})$Proof.

,$\xi\in S_{A\otimes I}(U\cross T)$.

I

This is asimple consequence of Proposition 3.2 (2) and (3).

With the help of Lemmas 6.7 and 6.8 we see immediately that conditions (01) andand(02) in Theorem 1.1 in the case whereare replaced with $U\cross\{1,2\}$ and GO,
respectively, coincide with the usual ones in Theorem 6.2.
$V$

$U$

7Characterization Theorems for Multi-Variable CaseLet us start with asingle CKS-space:$\mathcal{U}=\Gamma_{\alpha}(S_{A}(U))\subset\Gamma(L^{2}(U))\subset\Gamma_{\alpha}(S_{A}(U))^{*}=\mathcal{U}^{*}$

.

(16)

We are interested in multi-variable functions defined on $S_{A}(U)\cross\cdots\cross S_{A}(U)$ (n-times),in particular, of the forms:$F(\xi_{1}, \ldots, \xi_{m})=\langle\langle\Phi, \phi_{\xi_{1}}\otimes\cdots\otimes\phi_{\xi_{m}}\rangle\rangle$,$G(\xi_{1}, \ldots, \xi_{m}, \eta_{1}, \ldots, \eta_{n})=\langle\langle_{-}^{-}-(\phi_{\xi_{1}}\otimes\cdots\otimes\phi_{\xi_{m}}), \phi_{\eta_{1}}\otimes\cdots\otimes\phi_{\eta_{n}}\rangle\rangle$

,

(17)
(18)

. It is clear
, and
,
, , ,
where ,
G\^ateaux-entire.
that the functions defined in (17) and (18) are
The following results are immediate corollaries of Theorem 1.1 with the help of
Lemmas 6.7 and 6.8.
$\xi_{1}$$\ldots$$\xi_{m}$$\eta_{1}$$\ldots$$\eta_{n}\in S_{A}(U)$$\Phi$$\in(\mathcal{U}^{\otimes m})^{*-},--\in \mathcal{L}(\mathcal{U}^{\otimes m}, (\mathcal{U}^{\otimes n})^{*})$

is expressed in the form
Theorem 7.1 A G\^ateaux-entire function $F$ :(17) with
if and only if there exist constant numbers $C\geq 0$ and $p\geq 0$ suchthat
$S_{A}(U)^{m}arrow \mathrm{C}$$\Phi\in(\mathcal{U}^{\otimes m})^{*}$,$|F( \xi_{1}, \ldots, \xi_{m})|^{2}\leq C\prod_{j=1}^{m}G_{\alpha}(|\xi_{j}|_{p}^{2})$$\xi_{1}$,$\ldots$, $\xi_{m}\in S_{A}(U)$ .

197
Theorem 7.2 A(17) with cl\rangle cU’m-entire function F
C is expressed in the form
and only for any p 0 there exists C 0 such that

$Gc\ovalbox{\tt\small REJECT} teaux$$\ovalbox{\tt\small REJECT}$$i\ovalbox{\tt\small REJECT}$$i\ovalbox{\tt\small REJECT}$$5_{A}(U)^{m}\ovalbox{\tt\small REJECT}$$\ovalbox{\tt\small REJECT}$$\ovalbox{\tt\small REJECT}$$|F( \xi_{1,\ldots:}\xi_{m})|^{2}\leq C\prod_{j=1}^{m}G_{1/\alpha}(|\xi_{j}|_{-p}^{2})$,$\xi_{1}$,$\ldots$, $\xi_{m}\in S_{A}(U)$ .

Theorem 7.3 A G\^ateaux-entire function :is expressed in the form
$C\geq0$(18)
and $p\geq 0$ such that
if and only if there exist$G$$S_{A}(U)^{m+n}arrow \mathrm{C}$$with—\in \mathcal{L}(\mathcal{U}^{\otimes m}, (\mathcal{U}^{\otimes n})^{*})$$|G( \xi_{1}, \ldots, \xi_{m}, \eta_{1}, \ldots, \eta_{n})|^{2}\leq C\prod_{j=1}^{m}G_{\alpha}(|\xi_{j}|_{p}^{2})\prod_{k=1}^{n}G_{\alpha}(|\eta_{k}|_{p}^{2})$for$\xi_{1}$,$\ldots$,$\xi_{m}$,$\eta_{1}$,$\ldots$,, $\eta_{n}\in S_{A}(U)$ .

In that case, since
whenever $m’+n’=m+n$ .

$\mathcal{L}(\mathcal{U}^{\otimes m}, (\mathcal{U}^{\otimes n})^{*})\cong(\mathcal{U}^{\otimes(m+n)})^{*}$, we may choose an operator$\cup--\mathrm{f}\mathrm{r}\mathrm{o}\mathrm{m}$$\mathcal{L}(\mathcal{U}^{\otimes m’}, (\mathcal{U}^{\otimes n’})^{*})$

Theorem 7.4 A G\^ateaux-entire function :is expressed in the form(18) with
if and only if for any $p\geq 0$ there exist $C\geq 0$ and $q\geq 0$such that$G$$S_{A}(U)^{m+n}arrow \mathrm{C}$$—\in \mathcal{L}(\mathcal{U}^{\otimes m},\mathcal{U}^{\ovalbox{\tt\small REJECT}})$$|G( \xi_{1}, \ldots, \xi_{m}, \eta_{1,\ldots:}\eta_{n})|^{2}\leq C\prod_{j=1}^{m}G_{\alpha}(|\xi_{j}|_{p+q}^{2})\prod_{k=1}^{n}G_{1/\alpha}(|\eta_{k}|_{-p}^{2})$for$\xi_{1}$,$\ldots$,$\xi_{m}$,$\eta_{1}$,$\ldots$,, $\eta_{n}\in S_{A}(U)$ .

References[1] N. Asai, I. Kubo and H.-H. Kuo: General characterization theorems and intrinsic
topologies in white noise analysis, to appear in Hiroshima Math. J. 31 (2001).
[2] D. M. Chung, T. S. Chung and U. C. Ji: A simple proof of analytic characterization
theorem for operator symbols, Bull. Korean Math. Soc. 34 (1997), 421-436.
[3] D. M. Chung, U. C. Ji and N. Obata: Higher powers of quantum white noises interms of integral kernel operators, Infinite Dimen. Anal. Quantum Probab. Rel.
Top. 1(1998), 533-559.
[4] W. G. Cochran, H.-H. Kuo and A. Sengupta: A new class of white noise generalized
functions, Infinite Dimen. Anal. Quantum Prob. 1(1998), 43-67.

[5] R. Gannoun, R. Hachaichi, H. Ouerdiane and A. Rezgui: Un theoreme de dualite
entre espaces de fonctions holomorphes \‘a croissance exponentielle, J. Funct. Anal.
171 (2000), 1-14.
[6] 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.
[7] U. C. Ji and N. Obata: A role of Bargmann-Segal spaces in characterization andexpansion of operators on Fock space, preprint, 2000

198
[8] U. C. Ji, N. Obata and H. Ouerdiane: Analytic characterization of generalized Fockspace operators as twO-variable entire functions with growth condition, to appear in
Infinite Dimen. Anal. Quantum Probab. Rel. Top.
[9] I. Kubo and H.-H. Kuo: Simple proof of Hida distribution characterization theorem,
in “Exploring Stochastic Laws (A. V. Skorokhod and Yu. V. Borovskikh, Eds.),”pp. 243-250, VSP, Utrecht, 1995.
[10] I. Kubo, H.-H. Kuo and A. Sengupta: White noise analysis on a neat space of Hida
distributions, Infinite Dimen. Anal. Quantum Probab. Rel. Top. 2(1999), 315-335.
[11] I. Kubo and S. Takenaka: Calculus on Gaussian white noise I-IV, Proc. Japan
Acad. 56A (1980), 376-380; 411-416; 57A (1981), 433-437; 58A (1982), 186189.

[12] N. Obata: “White Noise Calculus and Fock Space,” Lect. Notes in Math. Vol. 1577,
Springer-Verlag, 1994.
[13] N. Obata: Inverse -transform, Wick product and overcompleteness of exponential
vectors, to appear in “Quantum Information IV (T. Hida and K. Saito, Eds.),”World Scientific.
[14] H. Ouerdiane: Fonctionnelles analytiques avec condition de croissance, et application \‘a l’analyse gaussienne, Japan. J. Math. 20 (1994), 187-198.
$S$[15] J. Potthoff and L. Streit: A characterization of Hida distributions, J. Funct. Anal.
101 (1991), 212-229