Controlled Frame for Operator in Hilbert c-Modules

Abstract

In this study, we will introduce a new concept, which is a controlled -operator frame for the space of all adjointable operators on a Hilbert -module which denoted , where is a -algebra. Also, we establish some results of the controlled -operator frame in . The presented results are new and of interest for people working in this area. Some illustrative examples are provided to advocate the usability of our results.

1. Introduction

Frame theory has a great revolution for recent years; this theory has several properties applicable in many fields of mathematics and engineering and plays a significant role in signal and image processing, which leads to many applications in informatics, medicine, and probability. Frame theory has been extended from Hilbert spaces to Hilbert -modules and has begun to be study widely and deeply. The basic idea was to consider module over -algebra instead of linear spaces and to allow the inner product to take values in the -algebra.

The concept of frames in Hilbert spaces has been introduced by Duffin and Schaeffer [1] in 1952 to study some deep problems in nonharmonic Fourier series. After the fundamental paper [2] by Gabor, frame theory began to be widely used, particularly, in the more specialized context of wavelet frames and Gabor frames [3]. Frames have been used in signal processing, image processing, data compression, and sampling theory.

In 2000, Frank and Larson [4] have extended the theory for the elements of -algebras and Hilbert -modules. Eventually, frames with -valued bounds in Hilbert -modules have been considered in [5].

The theory of frames has been generalized rapidly and there are various generalizations of frames in Hilbert spaces and Hilbert -modules.

In 2012, Gavruta [6] introduced the notion of -frames in Hilbert space to study the atomic systems with respect to a bounded linear operator .

Controlled frames in Hilbert spaces have been introduced in 2010 by Balazs et al. [7] to improve the numerical efficiency of iterative algorithms for inverting the frame operator, while the notion of controlled -frames in Hilbert spaces was introduced in 2015 by Nouri et al. [8].

Controlled frames in -modules were introduced by Kouchi and Rahimi [9], and the authors showed that they share many useful properties with their corresponding notions in a Hilbert space.

The notion of -operator frame for , which is a generalization of -frames in Hilbert -modules, is studied by Rossafi and Kabbaj [10].

Motivated by the above literature, we introduce the notion of controlled -operator frame for Hilbert -modules which is a generalization of -operator frame for .

2. Preliminaries

In the following, we briefly recall the definitions and basic properties of -algebra and Hilbert -modules. Our references for -algebras are [11, 12].

For a -algebra , if is positive, we write and denotes the set of all positive elements of .

Definition 1 (see [13]). Let be a unital -algebra and be a 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 is it is sesquilinear and positive definite and respects the module action. In other words,(i), for all , and if and only if (ii), for all and (iii), for all For we define the norm of by . If is complete with , it is called a Hilbert -module or a Hilbert -modules 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 .
denotes the range of the operator , and the set of all adjointable, positive, and invertible operators with bounded inverse will be denoted by .
The following lemmas will be used to prove our main results.

Lemma 1 (see [14]). Let and be two Hilbert -modules and . The following statements are equivalent:(i)is surjective.(ii)is bounded below with respect to norm, i.e., there issuch that(iii)is bounded below with respect to the inner product, i.e., there issuch that

Lemma 2 (see [13]). Let be a Hilbert -module and ; then, we have, for all ,

Theorem 1 (see [15]). Let be a Hilbert -module over a -algebra and let . If is closed, then the following statements are equivalent:(1)(2), for some(3)There existssuch that

Lemma 3 (see [16]). Let be a Hilbert -module. If is an invertible -linear map, then, for all , we have

Lemma 4 (see [5]). If is a -homomorphism between -algebras, then is increasing, that is, if , then .

3. Controlled -Operator Frame for

We begin this section with the following definitions.

Definition 2 (see [10]). Let . A sequence of adjointable operators on a Hilbert -module over a unital -algebra is called a -operator frame for if there exist two positive constants such thatThe numbers and are called lower and upper bound of the operator frame, respectively.

Definition 3 (see [17]). Let . A sequence of adjointable operators on a Hilbert -module over a unital -algebra is said to be a -controlled operator frame for if there exist two positive constants such that, for all , we haveThe numbers and are called lower and upper bounds of the -controlled operator frame, respectively.
Now, we define the notion of controlled -operator frame for .

Definition 4. Let and . A sequence of adjointable operators on a Hilbert -module over a unital -algebra is said to be a -controlled K-operator frame for if there exist two positive constants such thatThe numbers and are called lower and upper bound of the -controlled -operator frame, respectively.
If , the -controlled -operator frame is called -tight.
If , it is called a Parseval -controlled -operator frame.
If only upper inequality of (7) holds, then is called a -controlled -operator Bessel sequence for .

Example 3.4. Let , , and be a -controlled K-frame for .
Let such thatwhere .
From the definition of , there exist two constants such that, for all , we haveHence,soThen,Therefore,Since is a surjective operator, then, from Lemma 1, there exists , such thatThen,which shows that is a -controlled -operator frame for .

Proposition 1. Every -controlled operator frame for is a -controlled -operator frame for .

Proof. Let be a -controlled operator frame for with bounds and ; then, we have, for all ,From Lemma 2, we haveTherefore, is a -controlled -operator frame for with bounds and .

Proposition 2. Let be a -controlled K-operator frame for . If is a surjective operator, then is a -controlled operator frame for.

Proof. Let be a -controlled K-operator frame for with bounds A and B; hence,Since is surjective, then, from Lemma 1, there exists such thatUsing (18) and (19), we haveTherefore, is a -controlled operator frame for with bounds and .

Proposition 3. Let and be a K-operator frame for . Assume that and commutes between them and commutes with the operatorsand, for each. Then,is a-controlled-operator frame for.

Proof. Let be a -operator frame for ; then, there exist such thatOn the one hand, we have, for all ,Then,On the other hand, we have, for all ,From Lemma 1, there exists such thatFrom (23) and (24), we have, for all ,Therefore, is a -controlled -operator frame for .
From now, we suppose that and commutes between them and commutes with the operators for each . Let be a -controlled -operator Bessel sequence for . We define the operator byThe operator , called the analysis operator, is well defined and bounded. The adjoint operator of , called the synthesis operator, is defined byWe define the frame operator for byIt is clear to see that is a positive, bounded, and self-adjoint operator.

Theorem 2. Let be a -controlled operator Bessel sequence for . The following statements are equivalent:(1)is a-controlled-operator frame(2)There existssuch that(3), for some

Proof. : let be a -controlled -operator frame for with bounds and and frame operator ; then, we have, for all ,Therefore,soHence,: let such thatThis givesFrom Theorem 1, we havefor some .
: suppose thatfor some .
From Theorem 1, there exists such thatHence,Therefore, is a controlled K-operator frame for .

Proposition 4. Let and be a -controlled -operator frame for . Suppose that commutes with ,, and. Then,is a-controlled-operator frame for.

Proof. Suppose that is a -controlled K-operator frame with bounds and ; then, we have, for all ,Hence,soTherefore, is a -controlled -operator frame for with bounds and .

Theorem 3. Let and be a -controlled tight K-operator frame for with bound . Then, is a -controlled tight operator frame for with bound if and only if .

Proof. Let be a -controlled tight -operator frame for with bound ; then, we have, for all ,Assume that is a -controlled tight operator frame for with bound ; then, we have, for all ,Hence,soThen,Therefore,For the converse, assume thatThen,This givesOr is a -controlled tight K-operator frame for with bound , then we haveTherefore, is a -controlled tight operator frame for .

Corollary 1. Letandbe a sequence for. Then, those statements are true: Ifis a-controlled tight-operator frame for, thenis a-controlled tight-operator frame for Ifis a-controlled tight operator frame for, thenis a-controlled tight-operator frame for

Theorem 4. Let be a -controlled K-operator frame for with best frame bounds A and B. If is an adjointable and invertible operator such thatcommutes with, thenis a-controlled K-operator frame forwith best frame boundsandsatisfying the inequalities:

Proof. Let be a -controlled -operator frame for with best frame bounds A and B; then, for all , we haveAlso, we haveHence,Therefore, is a -controlled K-operator frame for with bounds and .
Now, let and be the best bounds of the -controlled -operator frame . Then,Since is a -controlled -operator frame for with frame bounds and , we have, for all ,Hence,Since and are the best frame bounds of -controlled -operator frame , we haveTherefore, inequality (53) follows from (57) and (60).

4. Tensor Product of Controlled -Operator Frame

Theorem 5. Let and be two Hilbert -modules over a unital -algebras and , respectively. Let and , and letbe a-controlled-operator frame forandbe a-controlled-operator frame forwith frame operatorsandand boundsand, respectively, whereand. Then,is a-controlled-operator frame for Hilbert-module, with frame operatorand boundsand.

Proof. According to the definitions of and contained in the previous theorem, we have, for all and ,Therefore,which givesThen,soThen,Therefore, is a -controlled -operator frame for Hilbert -module .
Moreover, we have, for all and ,By uniqueness of frame operator, the last equality is equal to , which ends the proof.

Theorem 6. Let and be two Hilbert -modules over a unital -algebras and , respectively. Let and , , and in, whereis an invertible operator. Leta sequence of adjointable operators inbe a-controlled-operator frame forwith lower and upper boundsand, respectively, and frame operator. Ifcommutes with,, and, thenis a-controlled-operator frame forwith boundsandand frame operator.

Proof. Let ; then, with inverse . Also, it is easy to show that the adjoint of is , and we have, for all ,which show that is bounded. Similarly, for , hence, is -linear, adjointable with adjoint .
So, for all , by Lemma 3, we haveFrom the definition of , we have, for all ,Then,On the one hand, we haveOn the other hand, we haveFinally, we obtainConcerning the frame operator, we havewhich allows us to find the requested result.

Theorem 7. Let and be two Hilbert -modules over a unital -algebras and , respectively. Let and , , and be in, whereis an invertible operator. Leta sequence of adjointable operators inbe a-controlled-operator frame forwith lower and upper boundsand, respectively, and frame operator. Ifcommutes with,, and, thenis a-controlled-operator frame forwith boundsandand frame operator.

Proof. It is similar to the proof of the previous theorem.

Theorem 8. Let and be two Hilbert -modules and let between -algebras be a -homomorphism andbe a map onsuch that, for all. Also, suppose thatis a-controlled-operator frame forwith frame operator. Ifis surjective and commutes with, and, thenis a-controlled-operator frame forwith frame operatorand.

Proof. Let ; since is surjective, then there exist such that . By the definition of -controlled -operator frame for , there exist such thatBy Lemma 4, we havewhich shows that is a -controlled -operator frame for .
Moreover, let and be the frame operators for and , respectively; then, we havewhich ends the proof.

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.