Journal of Mathematics

Journal of Mathematics / 2021 / Article

Research Article | Open Access

Volume 2021 |Article ID 5576534 | https://doi.org/10.1155/2021/5576534

Abdeslam Touri, Hatim Labrigui, Mohamed Rossafi, Samir Kabbaj, "Perturbation and Stability of Continuous Operator Frames in Hilbert -Modules", Journal of Mathematics, vol. 2021, Article ID 5576534, 15 pages, 2021. https://doi.org/10.1155/2021/5576534

Perturbation and Stability of Continuous Operator Frames in Hilbert -Modules

Academic Editor: Ali Jaballah
Received11 Jan 2021
Accepted01 Jun 2021
Published16 Jun 2021

Abstract

Frame theory has a great revolution in recent years. This theory has been extended from the Hilbert spaces to Hilbert -modules. In this paper, we consider the stability of continuous operator frame and continuous -operator frames in Hilbert -modules under perturbation, and we establish some properties.

1. Introduction and Preliminaries

The concept of frames in Hilbert spaces is a new theory which was introduced by Duffin and Schaeffer [1] in 1952 to study some deep problems in nonharmonic Fourier series. This theory was reintroduced and developed by Daubechieset al. [2].

In 1993, Aliet al. [3] introduced the concept of continuous frames in Hilbert spaces. Gabardo and Han in [4] called these kinds of frames, frames associated with measurable spaces.

In 2000, Frank and Larson [5] introduced the notion of frames in Hilbert -modules as a generalization of frames in Hilbert spaces. The theory of continuous frames has been generalized in Hilbert -modules. For more details, see [625].

The aim of this paper is to extend the results of Rossafi and Akhlidj [23], given for Hilbert -module in a discrete case.

In the following, we briefly recall the definitions and basic properties of -algebra and Hilbert -modules. Our references for -algebras are [26, 27]. For -algebra , if is positive, we write , and denotes the set of positive elements of .

Definition 1. (see [26]). Let be unital -algebra and be left -module, such that the linear structures of and are compatible. is a pre-Hilbert -module if is equipped with an -valued inner product , such that it is sesquilinear and positive definite and respects the module action. In the other words,(i), for all , and if and only if .(ii), for all and .(iii), for all .For we define . If is complete with , it is called a Hilbert -module or a Hilbert -module over .
For every in -algebra , we have and the -valued norm on is defined by , for all .
Let and be two Hilbert -modules, a map is said to be adjointable if there exists a map such that for all and .
We reserve the notation for the set of all adjointable operators from to and is abbreviated to .
The following lemmas will be used to prove our mains results.

Lemma 1 (see [28]). Let be a Hilbert -module. If , then

Lemma 2 (see [29]). Let and be two Hilbert -modules and . Then the following statements are equivalent:(i) is surjective.(ii) is bounded below with respect to norm, i.e., there is such that , for all .(iii) is bounded below with respect to the inner product, i.e., there is such that , for all .

Lemma 3 (see [30]). Let be a measure space, and are two Banach spaces, be a bounded linear operator and measurable function; then,

2. Characterisation of Continuous Operator Frame for

Let be a Banach space, a measure space, and be a measurable function. Integral of Banach-valued function has been defined by Bochner and others. Most properties of this integral are similar to those of the integral of real-valued functions [30, 31]. Since every -algebra and Hilbert -module are Banach spaces, we can use this integral and its properties.

Let be a measure space, and be two Hilbert -modules over a unital -algebra and is a family of submodules of . is the collection of all adjointable -linear maps from into .

We define the following:

For any and , the -valued inner product is defined by and the norm is defined by . In this case, is an Hilbert -module [32].

Definition 2. We call a continuous operator frame for if(a)for any , the mapping defined by is measurable(b)there is a pair of constants such that for any ,The constants and are called continuous operator frame bounds.
If , we call this continuous operator frame a continuous tight operator frame, and if , it is called a continuous Parseval operator frame.
If only the right-hand inequality of (4) is satisfied, we call the continuous Bessel operator frame for with Bessel bound .
The continuous frame operator of on is defined byThe continuous frame operator S is a bounded, positive, self-adjoint, and invertible.

Theorem 1. Let . is a continuous operator frame for if and only if there exist constants such that for any ,

3. Perturbation and Stability of Continuous Operator Frame for

Theorem 2. Let be a continuous operator frame for with bounds and . If is a continuous operator Bessel family with bound , then is a continuous operator frame for .

Proof. We just prove the case that is a continuous operator frame for .
On the one hand, for each , we haveHence,One the other hand, we haveThen,From (8) and (9), we getTherefore, is a continuous operator frame for .

Theorem 3. Let be a continuous operator frame for with bounds and and let . The following statements are equivalent:(i) is a continuous operator frame for .(ii)There exists a constant , such that for all x in , we have

Proof. Suppose that is a continuous operator frame for with bound and . Then for all , we haveIn the same way, we haveFor (12), we take .
Now we assume that (12) holds. For each , we haveFrom (12), we haveThen,Hence,Also, we haveFrom (12), we haveThen,So,From (18) and (22), we give thatTherefore, is a continuous operator frame for .

Theorem 4. Let , be a continuous operator frames for with bounds and and let be any scalars. If there exists a constant and some such thatthen is a continuous operator frame for and conversely.

Proof. For every , we haveHenceTherefore, is a continuous operator frame for .
For the converse, let be a continuous operator frame for with bounds , and let any .
Since is a continuous operator frame for with bounds and , then for any , we haveHence,Also, we haveThen,So,Then for , we havewhich ends the proof.

Theorem 5. For , let be continuous operator frames with bounds and and let .
Let be a bounded linear operator such thatIf there exists a constant such that for each and , we haveThen is a continuous operator frame for .

Proof. For all , we haveSince, for any , we haveThenHenceThereforeThis give that is a continuous operator frame for .

4. Characterisation of Continuous K-Operator Frames for

Definition 3. Let . A family of adjointable operators on a Hilbert -module is said to be a continuous K-operator frame for , if there exists two positive constants such thatThe numbers and are called, respectively, lower and upper bound of the continuous K-operator frame.
The continuous K-operator frame is called a -thight if:If , it is called a normalised tight continuous K-operator frame or a Parseval continuous K-operator frame. The continuous K-operator frame is standard if for every , the sum (40) converges in norm.

Remark 1. For any , every continuous operator frame is a continuous K-operator frame.
Indeed, for any , we haveLet be a continuous operator frame with bounds and , thenHence is a continuous K-operator frame with bounds and .
Let be a continuous K-operator for . We define the operatorThe operator is called the analysis operator of the continuous K-operator frame , and its adjoint is defined as follows:The operators is called the synthesis operator of the continuous K-operator frame .
By composing and , we obtain the operatorIt is easy to show that the operator is positive and self-adjoint.

Theorem 6. Let be a family of adjointable operators on a Hilbert -module . Assume that converges in norm for all . Then is a continuous K-operator frame for if and only if there exists two positive constants such that

Proof. Suppose that is a continuous K-operator frame.
From the definition of continuous K-operator frame, (47) holds.
Conversely, assume that (47) holds. The frame operator is positive and self-adjoint; thenWe have for anyUsing Lemma 2, there exist two constants such thatThis proves that is a continuous K-operator frame for .

5. Perturbation and Stability of Continuous K-Operator Frames for

Theorem 7. Let be a continuous K-operator frame for with bounds A and B, let and be two positively families. If there exist two constants such that for any , we have

Then is a continuous K-operator frame for .

Proof. For every , we haveThen,ThereforeHenceAlso, for all , we havethenHenceThusThereforeSo,HenceThis give that is a continuous K-operator frame for .

Theorem 8. Let be a continuous K-operator frame for with bounds and . Let and . If such that for all , we haveThen is a continuous K-operator frame with bounds and .

Proof. Let be a continuous K-operator frame with bounds and . Then for any , we have