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Touri, Abdeslam; Labrigui, Hatim; Rossafi, Mohamed; Kabbaj, Samir

doi:https://doi.org/10.1155/2021/5576534

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Abstract Introduction and Preliminaries Data Availability Conflicts of Interest References Copyright Related Articles

Research Article | Open Access

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

Perturbation and Stability of Continuous Operator Frames in Hilbert -Modules

Abdeslam Touri,¹Hatim Labrigui,¹Mohamed Rossafi,²and Samir Kabbaj¹

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 [6–25].

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 haveThereforeThusAlso, we haveHenceThereforeHence is a continuous K-operator frame with bounds and .

Corollary 1. Let be a continuous K-operator frame for with bounds and . Let and . If such thatthen is a continuous K-operator frame with bounds and .

Proof. The proof comes from the previous theorem.

Theorem 9. Let be a continuous K-operator frame for with bounds and . Let . If there exists such that for any , we haveThen is a continuous K-operator frame for . The converse is true for any surjective operator K such that in particular if K is co-isometry.

Proof. Assume that (71) holds.
On the one hand, we have for any x in On the other hand, we haveThenFrom (73) and (75), we obtainHence is a continuous K-operator frame for .
For the converse, if is a continuous K-operator frame for with bound and , and verify that, i.e, , then for every , we haveSimilarly we can obtainWe take , then (5.1) is verified.

Theorem 10. Let . For , let be a continuous K-operator frame for with bounds and , {} be any scalars. If there exists a constant and such thatthen is a continuous K-operator frame for . The converse is true for every co-isometric operator K.

Proof. For all , we haveHence for any , we haveThenThis gives that is a continuous K-operator frame for . For the converse, let K be a co-isometric operator on , let is a continuous K-operator frame for with bounds and and let for all , be a continuous K-operator frame for with bounds and . Then, for every , we haveThenAlso, we haveSince K is a co-isometric operator, thenSoTherefore, for , we have

Theorem 11. Let . For , let be a continuous K-operator frame for with bounds and , and let be any family. Let S: be a bounded linear operator such that , for some . If there exists a constant such thatthen is a continuous K-operator frame for .

Proof. For every , we haveWe have for anyHenceSoThereforeThusThis gives that is a continuous K-operator frame for .

Data Availability

No data were used to support this study.

Conflicts of Interest

On behalf of all authors, the corresponding author states that there are no conflicts of interest.

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Copyright

Copyright © 2021 Abdeslam Touri et al. This is an open access article distributed under the Creative Commons Attribution License, which permits unrestricted use, distribution, and reproduction in any medium, provided the original work is properly cited.

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