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).