Characterization Theorem of Generalized Operators

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ISSN Print: 2327-4352

DOI: 10.4236/jamp.2018.67120 Jul. 13, 2018 1434 Journal of Applied Mathematics and Physics

Characterization Theorem of Generalized

Operators

Mahmmoud Salih

*

, Sulieman Jomah

School of Mathematics and Statistics, Northwest Normal University, Lanzhou, China

Abstract

In this paper, by using the W-transform of an operator on white noise func-tionals, 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 in-troduced 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 opera-tor on white noise functionals. Obata [3] proved an analytic characterization theorem for symbols of operators on white noise functionals, which is an opera-tor version of the characterization theorem for white noise functionals (see, e.g.,

[4] [5] [6] [7]). Recently a Characterization theorem of operators of dis-crete-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 How to 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

Received: May 12, 2018 Accepted: July 10, 2018 Published: July 13, 2018

http://creativecommons.org/licenses/by/4.0/

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DOI: 10.4236/jamp.2018.67120 1435 Journal of Applied Mathematics and Physics

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

*_.

E H⊂ ⊂E

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

(

)

2 *_,

L E µ the Hilbert space of -valued L2_{-function on}_E*_{. The Winer}

-Itô-Segal theorem say that _{L E}2

(

*_,_µ

)

_{is unitary isomorphic to Boson Fock}

space Γ

( )

H_ _{. The isomorphism is a unique linear extension of the following}

correspondence between exponential functions and exponential vector:

2

, , 2

e 1, , , , , .

2! !

n

x

n

ξ ξ ξ ξ

ξ ξ

φ ₌ − _↔_ ξ ⊗ ⊗ _

 

  

If _φ_∈_{L E}2

(

*_,_µ

)

_and

( )

0 n n

f ∞₌ ∈Γ H_ _{are related through the}

Wiener-Itô-Segal isomorphism, we write

( )

~ fn

φ

for simplicity. It is then noted that

2 2

0 0

0 ! n , n n f

φ ∞

=

∑

_(2.1)

where

φ

0 is the L2-norm of φ∈L E2

(

*,µ

)

.

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

2 2 2

1 d d

A= + −t t we introduce a sequence of norms in H_ in such a way

that p ₀

p A

ξ = ξ _{. The number:}

1 1 1

0 1,

2

op Hs

A A

ρ − δ −

< = = < =

are frequently used. Suppose Ep is the Hilbert space obtained by completing E

with respect to the norm ._p. Then it is known that

*

lim _p, lim _p.

p p

E proj≅ _→∞E E ≅ind _→∞E−

The norm is naturally extended to the tensor product E⊗n_{and their}

complexification E⊗n

 . The canonical bilinear form .,. is also extended to a

-bilinear form on

( )

E⊗n *_×

( )

E ⊗n

  .

Let 0≤ <β 1 be a fixed number. For p≥0, define

( )

2

( )

2 1

,

0 ! n , ~ n .

p p

n n f f

β β

φ ∞ + φ

=

∑

_(2.2)

(3)

( )

E β =projlim_p→∞

( )

Ep β,

which becomes a countable Hilbert nuclear space. Next, we consider the dual spaces. For 0≤ <

β

1 and p≥0, define

( )

2

( )

2 1

,

0 ! n , ~ n .

p p

n n f f

β β

φ ∞ − φ

− − −

=

∑

(2.3)

Then .2_{− −}_p_, _β is a Hilberatian norm on _{L E}2

(

*_,_µ

)

_{and we denote by}

( )

E−p ₋_β the completion. The dual spaces of

( )

E β is given by

( )

*

( )

0

lim p p ,

p _p

E _β ind _→∞ E− ₋_β E− ₋_β ≥

≅ =



and we come to a complex Gelfand triple:

( )

_E _{L E}2

(

*_,

)

( )

_E *_.

β⊂

µ

⊂ β

Spaces

( )

E _β and

( )

E *_β 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

( ) ( )

E *_β× E _β_{will be denoted by}

.,. . Then

( ) ( )

*

( ) ( )

0

, ! n, n , ~ n , ~ n .

n n F f F E β f E β

φ ∞ φ

=

Φ =

∑

Φ ∈ ∈ _(2.4)

For ξ ∈E__{, define the renormalized exponential function} _{: e}.,ξ _:_by .,

0

1

: e : :. :, .

!

n n

ξ ∞ ⊗ _ξ⊗

=

∑

Moreover,

., ., , 2

: e ξ : e₌ ξ−ξ ξ .

It is a fact that _:_e.,ξ _:_{is a test function in}

_{( )}

_E

β for any ξ ∈E.

Definition 2.1. The S-transform of a generalized function Φ ∈

( )

E *_β _is

defined to be the function

( )

_{,: e}., _{: ,} _.

S_Φ _ξ _{= Φ} ξ _ξ_∈E

 (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 F S= Φ of Φ ∈

( )

E *_β satisfies the following conditions:

1) For any ξ_andη in E_, the function F z

(

ξ η+

)

is an entire function of z∈.

2) There exists nonnegative constant K, a and p such that

( )

exp 12 , .

p

F ξ ≤K _aξ −β_ ∀ ∈ξ E

  

(4)

and for any q satisfying the condition that

( )

1

2

2 2

e 1

1

q p Hs

a β _A

β

− − −

  _<

 ₋ 

  ,

the following inequality holds:

( )

1

1 ₂

2 2

,

2 1 e

1

q p

q _Hs

a

K A

β

β _β

− −

− − − −

 _ _ 

 

Φ ≤ _{− } _

 _ − _ 

 

Theorem 2.3. ([10]) Let F be a function on E_ satisfying the conditions: 1) For any ξ_andη in E, the function F z

(

ξ η+

)

is an entire function

of z∈.

2) There exists positive constant K, a and p such that

( )

exp 12 , .

p

F ξ K aξ +β ξ E

−

  ≤ _ _ ∀ ∈

  

Then there exists a unique

ϕ

∈

( )

Eq _β such that F S= Φ for any q∈

[

0,∞

)

satisfying the condition

( )

1

2

2 2

e 1

1

q p Hs

a β _A

β

+ − −

  _<

 ₊ 

 

and

( )

1

1 ₂

2 2

,

2

1 e .

1

q p

q _Hs

a

K A

β

ϕ

β

− +

− −

 _ _ 

 

≤ _{− } _

 _ + _ 

 

3. White Noise Operator

Let L E

(

( ) ( )

_β, E *_β

)

(resp. L E

(

( ) ( )

_β, E _β

)

) denote the space of all continuous linear operator from

( )

E _β into

( )

E *_β (resp.

( )

E _β). In this section, we shall prove a characterization theorem for an operator Ξ ∈L E

(

( ) ( )

_β, E *_β

)

and for an operator Ξ ∈L E

(

( ) ( )

_β, E _β

)

_.

The W-transform of an operator Ξ ∈L E

(

( ) ( )

_β, E *_β

)

is defined to be an

( )

*

E _β-valued function on E_ defined by

( )

, .

WΞ ξ = Ξϕ ξξ ∈E (3.1)

Note that the W-transform is injective and that for any φ∈

( )

E _β and

, E

ξ η ∈ _, we have _W_Ξ

(

_z_{ξ η φ}₊

)

_, ₌_S

( )

_Ξ*_φ

(

_z_{ξ η}₊

)

_,_z_∈__{, where}_Ξ*_{is the}

adjoint operator of Ξ_,_i.e._, _Ξ*_{is the continuous linear operator from}

_{( )}

_E β

into

( )

E *_β such that

( )

*

, , , , E _β.

φ ψ

ψ φ

φ ψ

Ξ = Ξ ∈

It follows from Theorem 2.2 that the function z→ WΞ

(

zξ η φ+

)

, is an entire function on .

We note that there exist p≥0 and K such that

( )

, , , .

p β K pβ E β

φ _{− −} φ φ

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Then, we have the following growth condition

2 1

, exp , ,

p p

W K a β E

β ξ − ξ

− −

 

Ξ ≤ _ _ ∈

   (3.2)

where K C₌ 2β_and _a_{= −}

(

₁ _β

)

2 1₁β₋_β−

Theorem 4.1. Let G be an

( )

E *_β-valued function on E_. Then there exists a continuous operator Ξ ∈L E

(

( ) ( )

_β, E *_β

)

_{such that}_G_{is the W-transform of} Ξ

if and only if G satisfies the following conditions:

1) For each ξ η ∈, E__{, and} φ∈

( )

E _β the function z→ G z

(

ξ η φ+

)

, is an entire function on .

2) There exist nonnegative constant K, a and p such that

( )

12

, exp p , .

p

G K a β E

β ξ − ξ

− −

 

Ξ ≤ _ _ ∀ ∈

  

Proof. In case of β =0 the proof is given in [1], the proof for general case

0≤ <β 1 is a simple modification. In fact, the first assertion was shown above. Conversely, suppose G is an

( )

E *_β-valued function on E_ satisfying (1) and (2), we need only to prove the existence of Ξ, fix an arbitrary φ∈

( )

E _β. Define a -valued function Fφ by

( )

, , .

Fφ ξ = G ξ φ ξ∈E

Then Fφ satisfies (1) and (2) in Theorem 2.2, clearly for any ξ η ∈, E, the

function Fφ

( )

ξ = G

( )

ξ φ, of z∈ is holomorphic on  and we have ,

for ξ ∈E_

( )

(

)

12

, ,

, q q exp p .

q

F_φ G K a β

β β

β

ξ φ φ ξ −

− −

 

≤ ≤ _ _

 

Hence, by Theorem 2.2, there exists a unique Φ ∈φ

( )

E *β such that

( )

, , .

SΦφ ξ =Fφ ξ = G ξ φ ξ∈E

Moreover, for any r p> with

( )

1

2

2 2 ₁

1

r p Hs

a

e A

β

− − −

  _<

 ₋ 

 

( )

1 2 1

2 2

, ,

2

1 e .

1

r p q

r Hs

a

K A

β

φ β β _β

− −

− − − −

 _ _ 

 

Φ ≤ Φ _{− } _

 _ − _ 

  (3.3)

Hence, the operator φ→ Φφ is continuous linear operator from

( )

Eq _β into

( )

* r

E _β, and we obtain Ξ ∈L E

(

( ) ( )

_β, E *_β

)

_with Ξϕξ =G

_{( )}

ξ .

Definition 4.2. For Ξ ∈L E

(

( ) ( )

_β, E *_β

)

_{a function on}E_×E__{is defined}

by

( )

ˆ ,

ξ η

φ φ

ξ, η , ,

ξ η

E

Ξ = Ξ ∈ __(3.4)

is called the symbol of Ξ_.

Corollary 4.3. Suppose that a -valued function on E_×E__{satisfies the}

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1) For each ξ ξ η, ,′ _andη′ in E_, the function

(

z w,

)

→F z

(

ξ ξ η η+ ′,w + ′

)

is an entire function on _{ }× _.

2) There exist p≥0,a>0 and K>0 such that

(

,

)

exp 12 12 , , .

p p

F ξ η ≤K a_ξ −β +η −β_ ξ η∈E

  

Then there exists a unique Ξ ∈L E

(

( ) ( )

_β, E *_β

)

such that F is the symbol of Ξ_.

The proof given in [1, 8, p.91] for case of β =0 is adjust to the general case

0≤ <β 1, see [7].

The W-transform of an operator Ξ ∈L E

(

( ) ( )

_β, E _β

)

is defined to be an

( )

E _β-valued function on E_ defined by

( )

, .

WΞ ξ = Ξϕ ξξ ∈E (3.5)

Then for any Φ ∈

( )

E *_β_and ξ η ∈, E__{, we see that}

(

)

( )

*

(

)

,W zξ η S zξ η , z

Φ + = Ξ Φ + ∈,

therefore z→ Φ,W z

(

ξ η+

)

is holomorphic on . Moreover, note that for each q≥0 there exist p≥0 and K>0 such that

( )

, , , .

qβ K pβ E β

φ φ φ

Ξ ≤ ∈

In particular, for all ξ ∈E_

2 1

, exp , .

q p

W K a β E

β ξ + ξ

 

Ξ ≤ _ _ ∈

   (3.6)

Theorem 4.4. Let G be an

( )

E _β-valued function on E_, satisfying the following conditions:

1) For each ξ η ∈, E_ _{, and for any}

φ

∈

( )

E *_β _{the function}

(

)

,

z→ φ G zξ η+ is an entire function on .

2) For any q≥0_{, there exist} p≥0,a>0_and K>0_{such that}

( )

12

, exp p , .

q

G K a β E

β ξ + ξ

  Ξ ≤ _ _ ∀ ∈

  

Then there exists a continuous operator Ξ ∈L E

(

( ) ( )

_β, E _β

)

such that G is the W-transform of Ξ_.

Proof. The proof is similar to the proof of Theorem 4.1. So, we shall prove the existence of Ξ_{. Fix an arbitrary}Φ ∈

( )

E *_β_{. Define a}-valued function FΦ

by

( )

,

( )

, .

F_Φ ξ = Φ G ξ ξ∈E_

Clearly,

F

Φ satisfies conditions (1) and (2) in the Theorem 2.3. Hence, by

Theorem 2.3, there exists a unique Ψ ∈Φ

( )

E *_β such that

( )

,

( )

, .

SΨΦ ξ = ΦG ξ ξ∈E

Moreover, for any r p> with

( )

1

2

2 2 ₁

1

r p Hs

a

e A

β

+ − −

  _<  ₊ 

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DOI: 10.4236/jamp.2018.67120 1440 Journal of Applied Mathematics and Physics ( )

1 2 1

2 2

, ,

2

1 e .

1

r p

r q _Hs

a

K A

β

β β _β

− +

− − Φ − − − −

 _ _ 

 

Ψ ≤ Φ _ _{− } _ _

+  

  (3.2)

Therefore, the operator Φ → ΨΦ is continuous linear operator from

( )

*

E _β

into

( )

E *_β . Now, let Ξ _{be the adjoint of this operator, then}

( ) ( )

(

,

)

L E _β E _β

Ξ ∈ _{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

{ }

n=1

∞

Ξ and Ξ _{be in}L E

(

( ) ( )

_β, E *_β

)

. Let

,

n n

G W= Ξ n∈_ and G W= Ξ . Then Ξn converges to Ξ strongly in

( ) ( )

(

*

)

,

L E _β E _β if and only if the following conditions are satisfied: 1) Gn

( )

ξ converges to G

( )

ξ in

( )

*

E _β for each ξ ∈E__.

2) There exist q≥0,p≥0,K>0 and a>0 such that

( )

12

, exp , , .

n q p

G K a β E n

β

ξ ξ − ξ

− −

 

≤ _ _ ∈ ∈

   

Proof. Suppose that Ξn converges to Ξ strongly in

(

( ) ( )

)

*

,

L E _β E _β . Then for each φ∈

( )

E _β, Ξnφ converges to Ξφ in

( )

*

E _β. Clearly (1) is satisfied. To prove (2), we consider

( )

{

}

, ,sup , .

q k _n n q

X β = ∈φ E β _∈ Ξφ − −β ≤k

 (4.1)

Then we have

( )

E β =



q, ,βk∈Xq, ,βk. Since

( )

E β is a Frécht space, by the

Baire’s category theorem there exist q and k in  such that Xq, ,βk contains an

open set of

( )

E _β. So we can see that there exist p∈ and ε >o_{such that}

( )

{

φ∈ E β;φ p,β <ε

}

⊂Xq k, ,β .

Then for any φ∈

( )

E _β, we have

, ,

n q p

k

β β

φ φ

ε

− −

Ξ ≤

′

for all n∈, where 0<ε′<ε. In particular, we have

( )

2 1

, , , e .

p

a

n _q n _q k _p k

G ξ _β ϕξ _β _ε ϕξ _β _ε ξ β

−

− − = Ξ − − ≤ _′ ≤ _′ (4.2) This completes the proof of the first assertion.

Conversely, assume that

{ }

Gn satisfies the given conditions. Then by (1), for

each ξ ∈E__andψ∈

_{( )}

E _β,

, , .

n

ϕ ψ

ξ

ϕ ψ

ξ

Ξ → Ξ

Since the linear span of

{

ϕ ξξ; ∈E_

}

is dense in

( )

E β, it follows from (2)

and Theorem 4.1 for any φ ψ, ∈

( )

E _β , Ξnφ ψ, converges to Ξφ ψ, .

This means that for any φ∈

( )

E _β, Ξnφ converges to Ξφ weakly in

( )

*

(8)

Hence, for any φ∈

( )

E _β , Ξnφ converges to Ξφ strongly in

( )

*

E _β. This completes the proof.

Corollary 4.2. let

{ }

n1

∞ =

Ξ and Ξ_{be in}L E

(

( ) ( )

_β, E *_β

)

. Let Fn= Ξˆ ,n n∈

and F= Ξˆ . Then Ξn converges to Ξ strongly in

(

( ) ( )

)

*

,

L E _β E _β if and only if the following conditions are satisfied:

1) For each ξ η ∈, E_, Fn

(

ξ η,

)

converges to F

(

ξ η,

)

.

2) There exist p≥0,K>0 and a>0 such that

(

,

)

exp 12 12 , , , .

n p p

F ξ η ≤K a_ξ −β +η −β_ ξ η∈E n∈

   

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 ξ η ∈, E__{and for} n∈,

( )( )

(

)

2 2

1 1

, ea p ea p .

n n

SG ξ η = F ξ η ≤K ξ −β η −β

Hence by Theorem 2.2, we have for any q p> with

( )

1

2

2 2

e 1

1

q p Hs

a β _A

β

− − −

  _<

 ₋ 

  ,

( )

2 1

1

1 ₂

2 2

,

2

e 1 e , , .

1

p

a q p

n _q a _Hs

G K β A E n

β ξ

β

ξ ξ

β

− − −

− − − −

 _ _ 

 

≤ − _ _ ∈ ∈

 _ − _ 

   

On the other hand, using (1) we can show that for ξ ∈E_ _,

( )

,

( )

,

n

G ξ φ → G ξ φ for all φ∈

( )

E _β. Hence conditions (1) and (2) are satisfied.

Theorem 4.3. let

{ }

n1

∞ =

Ξ and Ξ _{be in}L E

(

( ) ( )

_β, E _β

)

. Let

,

n n

G W= Ξ n∈_ _and G W= Ξ . Then Ξn converges to Ξ strongly in

( ) ( )

(

,

)

L E _β E _β if and only if the following conditions are satisfied: 1) For each ξ ∈E__, G_n

_{( )}

ξ converges to G

( )

ξ in

( )

E _β. 2) For each q≥0, there exist p>0,K>0,a>0 such that

( )

12

, exp , , .

n q p

G K a β E n

β

ξ ≤ _ξ + _ ξ∈ ∈

   

Proof. Suppose that Ξn converges to Ξ ∈L E

(

( ) ( )

β, E β

)

strongly in

( ) ( )

(

,

)

L E _β E _β . Then for any φ∈

( )

E _β, Ξnφ converges to Ξφ strongly in

( )

E _β. Hence (1) is obvious. To prove (2), given q≥0, we consider

( )

{

;sup ,

}

.

k _n n q

Y φ E _β φ _β k

∈

= ∈ Ξ ≤

 (4.3)

Then Yk is closed and

( )

E β =



k∈Yk . Hence by using the similar

arguments of the proof of Theorem 4.1, we can prove (2).

Conversely, assume that

{ }

Gn satisfies conditions (1) and (2). let q≥0,

then by (1), we have

,

lim n _q 0, .

(9)

Hence, by using (2) and Theorem 4.4, we can prove that for any φ∈

( )

E _β

,

lim n q 0.

n→∞ Ξ − Ξφ φ β =

This completes the proof. Corollary 4.4. let

{ }

_n∞₁

=

Ξ and Ξ be in L E

(

( ) ( )

_β, E _β

)

. Let Fn= Ξˆ ,n n∈

and F= Ξˆ . Then Ξn converges to Ξ strongly in L E

(

( ) ( )

β, E β

)

if and

only if the following conditions are satisfied:

1) For each ξ η ∈, E__, F_n

₍

ξ η,

₎

converges to F

(

ξ η,

)

.

2) For each q≥0 and a>0, there exist p q K≥ , 0> such that

(

,

)

exp 12 12 , , , .

n p q

F ξ η K a ξ +β η +β ξ η E n

−

 

≤ _ + _ ∈ ∈

   

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).

References

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