Journal of Applied Mathematics and Physics
Vol.06 No.07(2018), Article ID:85959,9 pages
10.4236/jamp.2018.67120
Characterization Theorem of Generalized Operators
Mahmmoud Salih*, Sulieman Jomah
School of Mathematics and Statistics, Northwest Normal University, Lanzhou, China
Copyright © 2018 by authors and Scientific Research Publishing Inc.
This work is licensed under the Creative Commons Attribution International License (CC BY 4.0).
http://creativecommons.org/licenses/by/4.0/
Received: May 12, 2018; Accepted: July 10, 2018; Published: July 13, 2018
ABSTRACT
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.
Keywords:
White Noise Functionals, W-Transform, S-Transform, Characterization Theorem
1. 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 operators on white noise functionals in term of their W-transform. Finally in Section 4, convergence of operators is discussed.
2. Preliminary Result on White Noise
We start with the real Gelfand triple
The norm of H is denoted by and since compatible the real inner product of H and the canonical bilinear form on are denoted by the same symbol . Suppose is the standard Gaussian measure on and the Hilbert space of -valued L2-function on . The Winer -Itô-Segal theorem say that is unitary isomorphic to Boson Fock space . The isomorphism is a unique linear extension of the following correspondence between exponential functions and exponential vector:
If and are related through the Wiener-Itô-Segal isomorphism, we write
for simplicity. It is then noted that
(2.1)
where is the L2-norm of .
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 we introduce a sequence of norms in in such a way that . The number:
are frequently used. Suppose is the Hilbert space obtained by completing E with respect to the norm . Then it is known that
The norm is naturally extended to the tensor product and their complexification . The canonical bilinear form is also extended to a
-bilinear form on .
Let be a fixed number. For , define
(2.2)
For each , becomes a Hilbert space. We put
which becomes a countable Hilbert nuclear space. Next, we consider the dual spaces. For and , define
(2.3)
Then is a Hilberatian norm on and we denote by the completion. The dual spaces of is given by
and we come to a complex Gelfand triple:
Spaces and are called spaces of test functions and generalized functions, respectively. The construction of these spaces is due to Kondaratiev and Streit [9] . The canonical bilinear form on will be denoted by . Then
(2.4)
For , define the renormalized exponential function by
Moreover,
It is a fact that is a test function in for any .
Definition 2.1. The S-transform of a generalized function is defined to be the function
(2.5)
A fundamental theorem in white noise analysis is the Kondratiev-Streit characterization theorem [9] (see also [10] ).
Theorem 2.2. ( [10] ) The S-transform of satisfies the following conditions:
1) For any and in , the function is an entire function of .
2) There exists nonnegative constant K, a and p such that
Conversely, suppose a -valued function F defined on satisfies the above two conditions. Then there exists a unique such that and for any q satisfying the condition that
,
the following inequality holds:
Theorem 2.3. ( [10] ) Let F be a function on satisfying the conditions:
1) For any and in , the function is an entire function of .
2) There exists positive constant K, a and p such that
Then there exists a unique such that for any satisfying the condition
and
3. White Noise Operator
Let (resp. ) denote the space of all continuous linear operator from into (resp. ). In this section, we shall prove a characterization theorem for an operator and for an operator .
The W-transform of an operator is defined to be an -valued function on defined by
(3.1)
Note that the W-transform is injective and that for any and , we have , where is the adjoint operator of , i.e., is the continuous linear operator from into such that
It follows from Theorem 2.2 that the function is an entire function on .
We note that there exist and K such that
Then, we have the following growth condition
(3.2)
where and
Theorem 4.1. Let G be an -valued function on . Then there exists a continuous operator such that G is the W-transform of if and only if G satisfies the following conditions:
1) For each , and the function is an entire function on .
2) There exist nonnegative constant K, a and p such that
Proof. In case of the proof is given in [1] , the proof for general case is a simple modification. In fact, the first assertion was shown above. Conversely, suppose G is an -valued function on satisfying (1) and (2), we need only to prove the existence of , fix an arbitrary . Define a -valued function by
Then satisfies (1) and (2) in Theorem 2.2, clearly for any , the function of is holomorphic on and we have , for
Hence, by Theorem 2.2, there exists a unique such that
Moreover, for any with
(3.3)
Hence, the operator is continuous linear operator from into , and we obtain with .
Definition 4.2. For a function on is defined by
(3.4)
is called the symbol of .
Corollary 4.3. Suppose that a -valued function on satisfies the following condition:
1) For each and in , the function is an entire function on .
2) There exist and such that
Then there exists a unique such that F is the symbol of .
The proof given in [1, 8, p.91] for case of is adjust to the general case , see [7] .
The W-transform of an operator is defined to be an -valued function on defined by
(3.5)
Then for any and , we see that
,
therefore is holomorphic on . Moreover, note that for each there exist and such that
In particular, for all
(3.6)
Theorem 4.4. Let G be an -valued function on , satisfying the following conditions:
1) For each , and for any the function is an entire function on .
2) For any , there exist and such that
Then there exists a continuous operator such that G is the W-transform of .
Proof. The proof is similar to the proof of Theorem 4.1. So, we shall prove the existence of . Fix an arbitrary . Define a -valued function by
Clearly, satisfies conditions (1) and (2) in the Theorem 2.3. Hence, by Theorem 2.3, there exists a unique such that
Moreover, for any with
(3.2)
Therefore, the operator is continuous linear operator from into . Now, let be the adjoint of this operator, then as desired.
4. Convergence of Operator
The convergence of operator sequences is rephrased in terms of convergence of W-transform and symbol.
Theorem 4.1. Let and be in . Let and . Then converges to strongly in if and only if the following conditions are satisfied:
1) converges to in for each .
2) There exist and such that
Proof. Suppose that converges to strongly in . Then for each , converges to in . Clearly (1) is satisfied. To prove (2), we consider
(4.1)
Then we have . Since is a Frécht space, by the
Baire’s category theorem there exist q and k in such that contains an open set of . So we can see that there exist and such that
.
Then for any , we have
for all , where . In particular, we have
(4.2)
This completes the proof of the first assertion.
Conversely, assume that satisfies the given conditions. Then by (1), for each and ,
Since the linear span of is dense in , it follows from (2) and Theorem 4.1 for any , converges to . This means that for any , converges to weakly in . Hence, for any , converges to strongly in . This completes the proof.
Corollary 4.2. let and be in . Let and . Then converges to strongly in if and only if the following conditions are satisfied:
1) For each , converges to .
2) There exist and such that
Proof. To prove the Corollary, it suffices to prove that (1) and (2) in Theorem 4.1 are equivalent to (1) and (2). Now assume that (1) and (2) are satisfied. Using (2), we can see that for and for ,
Hence by Theorem 2.2, we have for any with
,
On the other hand, using (1) we can show that for , for all . Hence conditions (1) and (2) are satisfied.
Theorem 4.3. let and be in . Let and . Then converges to strongly in if and only if the following conditions are satisfied:
1) For each , converges to in .
2) For each , there exist such that
Proof. Suppose that converges to strongly in . Then for any , converges to strongly in . Hence (1) is obvious. To prove (2), given , we consider
(4.3)
Then is closed and . Hence by using the similar arguments of the proof of Theorem 4.1, we can prove (2).
Conversely, assume that satisfies conditions (1) and (2). let , then by (1), we have
(4.4)
Hence, by using (2) and Theorem 4.4, we can prove that for any
This completes the proof.
Corollary 4.4. let and be in . Let and . Then converges to strongly in if and only if the following conditions are satisfied:
1) For each , converges to .
2) For each and , there exist such that
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).
Cite this paper
Salih, M. and Jomah, S. (2018) Characterization Theorem of Generalized Operators. Journal of Applied Mathematics and Physics, 6, 1434-1442. https://doi.org/10.4236/jamp.2018.67120
References
- 1. Chung, D.M., Chung, T.S. and Cig Ji., U. (1999) A Characterization Theorem for Operators on White Noise Functionals. Journal of the Mathematical Society of Japan, 51, 437-447. https://doi.org/10.2969/jmsj/05120437
- 2. Hida, T. (1975) Analysis of Brownian Functionals. Carleton Mathematical Lecture Notes 13, Carleton University, Ottawa.
- 3. Obata, N. (1993) An Analytic Characterization of Symbols of Operators on White Noise Functionals. Journal of the Mathematical Society of Japan, 45, 421-445. https://doi.org/10.2969/jmsj/04530421
- 4. Kuo, H.H., Potthoff, J. and Streit, L. (1990) A Characterization of White Noise Test Functionals. Nagoya Mathematical Journal, 121, 185-194. https://doi.org/10.1017/S0027763000003469
- 5. Potthoff, J. and Streit, L. (1991) A Characterization of Hida Distributions. Journal of Functional Analysis, 101, 212-229. https://doi.org/10.1016/0022-1236(91)90156-Y
- 6. Obata, N. (1994) White Noise Calculus and Fock Space. Lecture Notes in Mathematics. Springer-Verlag, Berlin, 1577. https://doi.org/10.1007/BFb0073952
- 7. Hida, T., Kuo, H.H., Potthoff, J. and Streit, L. (1995) White Noise: An Infinite Dimensional Calculus. Kluwer Academic Publishers, Dordrecht.
- 8. Wang, C. and Chen, J. (2016) A Characterization of Operators on Functionals of Discrete-Time Normal Martingale. Stochastic Analysis and Applications, 35, 305-316. https://doi.org/10.1080/07362994.2016.1248779
- 9. Kondratiev, Y.G. and Streit, L. (1993) Spaces of White Noise Distributions: Constructions, Descriptions, Applications I. Reports on Mathematical Physics, 33, 341-366. https://doi.org/10.1016/0034-4877(93)90003-W
- 10. Kuo, H.H. (1996) White Noise Distribution Theory. CRC Press, Boca Raton.