Characterization Theorem of Generalized Operators

In this paper, by using the W-transform of an operator on white noise functionals, we establish a general characterization theorem for operators on white noise functionals in term of growth condition. We also discuss convergence of operator sequences.


Introduction
The main purpose of the present paper is to obtain the characterization theorem for operators on white noise functionals in term of their W-transform was introduced by authors in [1], and give a criterion for the convergence of operators on white noise functionals in term of their W-transform.In [1] the authors deal with the standard Hida-Kubo-Takenka space, in this paper we deal with the Kondrateiv-Streit space, which is more suitable for our purpose.
On the other hand, the white noise calculus (or analysis) was launched out by Hida [2] in the Gaussian case, with his celebrated lecture notes.The concept of the symbol of an operator is of fundamental importance in the theory of operator on white noise functionals.Obata [3] proved an analytic characterization theorem for symbols of operators on white noise functionals, which is an operator version of the characterization theorem for white noise functionals (see, e.g., [4] [5] [6] [7]).Recently a Characterization theorem of operators of discrete-time normal martingales, was established in [8].This paper is organized as follows.Section 2, is dedicated to a quick review of white noise functionals.In Section 3, we prove the characterization theorems for The norm of H is denoted by 0 .and since compatible the real inner product of H and the canonical bilinear form on * E E × are denoted by the same symbol .,. .Suppose µ is the standard Gaussian measure on * E and ( ) L E µ the Hilbert space of  -valued L 2 -function on * E .The Winer -Itô-Segal theorem say that The isomorphism is a unique linear extension of the following correspondence between exponential functions and exponential vector: In order to introduce white noise distribution, we need a particular family of seminorms defining the topology of E. By means of the differential operator The norm is naturally extended to the tensor product n E ⊗ and their complexification n E ⊗  .The canonical bilinear form .,. is also extended to a  -bilinear form on ( ) ( )   the completion.The dual spaces of ( )  and we come to a complex Gelfand triple: For E ξ ∈  , define the renormalized exponential function ,: e : , .
A fundamental theorem in white noise analysis is the Kondratiev-Streit characterization theorem [9] (see also [10]).
satisfies the following conditions: 1) For any ξ and η in E  , the function ( ) 2) There exists nonnegative constant K, a and p such that ( ) Conversely, suppose a  -valued function F defined on E  satisfies the above two conditions.Then there exists a unique ( ) and for any q satisfying the condition that ( ) the following inequality holds: ( ) ]) Let F be a function on E  satisfying the conditions: 1) For any ξ and η in E  , the function ( ) 2) exists positive constant K, a and p such that ( ) Then there a unique ) denote the space of all continuous linear operator from ( ) In this section, we shall prove a characterization theorem for an operator Ξ ∈ and for an operator The W-transform of an operator Note that the W-transform is injective and that for any ( ) Ξ is the continuous linear operator from ( ) We note that there exist 0 p ≥ and K such that ( ) ,   , , .
Then, we have the following growth condition  ( ) Ξ ∈ such that G is the W-transform of Ξ if and only if G satisfies the following conditions: 1) For each , E ξ η ∈  , and ( ) 2) There exist nonnegative constant K, a and p such that ( ) In case of 0 β = the proof is given in [1], the proof for general case 0 1 β ≤ < is a simple modification.In fact, the first assertion was shown above.
Then there exists a unique The proof given in [1, 8, p.91] for case of 0 β = is adjust to the general case 0 1 β ≤ < , see [7].
The W-transform of an operator Ξ ∈ is defined to be an ( ) Then for any ( ) and , E ξ η ∈  , we see that + is holomorphic on  .Moreover, note that for each 0 q ≥ there exist 0 p ≥ and 0 K > such that ( ) ,   , , .
In particular, for all 2) For any 0 q ≥ , there exist 0, 0 p a ≥ > and 0 K > such that ( ) Then there exists a continuous operator Proof.The proof is similar to the proof of Theorem 4.1.So, we shall prove the existence of Ξ .Fix an arbitrary ( ) Clearly, Φ F satisfies conditions (1) and (2) in the Theorem 2.3.Hence, by Theorem 2.3, there exists a unique ( ) Moreover, for any r p > with ( ) Therefore, the operator Φ Φ → Ψ is continuous linear operator from ( )

Convergence of Operator
The convergence of operator sequences is rephrased in terms of convergence of W-transform and symbol.
and Ξ be in if and only if the following conditions are satisfied: 2) There exist 0, 0, 0 To prove ( Then we have ( ) E β is a Frécht space, by the Baire's category theorem there exist q and k in  such that , , q k X β contains an open set of ( ) E β .So we can see that there exist p ∈  and o ε > such that ( ) { } , , , ; Then for any ( ) for all n ∈  , where 0 ε ε ′ < < .In particular, we have ( ) This completes the proof of the first assertion.
Conversely, assume that { } n G satisfies the given conditions.Then by (1), for each E ξ ∈  and ( ) Since the linear span of { }

= ≤
Hence by Theorem 2.2, we have for any q p > with ( ) On the other hand, using (1) we can show that for E if and only if the following conditions are satisfied: Then for any ( ) Ξ strongly in ( ) Hence (1) is obvious.To prove (2), given 0 q ≥ , we consider  2) For each 0 q ≥ and 0 a > , there exist , 0 p q K ≥ > such that ( ) and we denote by function in ( )E β for any E ξ ∈  .Definition 2.1.The S-transform of a generalized function

4 )
is called the symbol of Ξ .Corollary 4.3.Suppose that a  -valued function on E E ×   satisfies the following condition: 1) For each , , ξ ξ η ′ and η′ in E  , the function (

. 4 )
Hence by using the similar arguments of the proof of Theorem 4.1, we can prove (2).Conversely, assume that { } n G satisfies conditions (1) and (2).let 0 q ≥ , then by (1), we have , Journal of Applied Mathematics and Physics Hence, by using (2) and Theorem 4.4, we can prove that for any Proof.The proof is straightforward by Corollary 4.2.We can prove that (1) and (2) in Theorem 4.3 are equivalent to (1) and (2). 