Controlled -Operator Frames on Hilbert -Modules

Abstract

In this paper, we study the concept of controlled -operator frames for the space of all adjointable operators on a Hilbert -module H. Also, we discuss characterizations of controlled -operator frames and we give some properties. Some illustrative examples are provided to advocate the usability of our results.

1. Introduction and Preliminaries

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 Daubechies et al., 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.

Controlled frames in Hilbert spaces have been introduced by Balazs [4] to improve the numerical efficiency of iterative algorithms for inverting the frame operator.

Controlled frames in Hilbert -modules were introduced by Rashidi and Rahimi [5], and the authors showed that they share many useful properties with their corresponding notions in a Hilbert space. -operator frames for has been studied by Rossafi and Kabbaj [6]. For more details, see [7–9].

In this paper, we introduce the notion of controlled -operator frames for , where is a Hilbert -modules.

Let be a countable index set. In this section, we briefly recall the definitions and basic properties of -algebra, Hilbert -modules, frame, and operator frame in Hilbert -modules. For information about frames in Hilbert spaces, we refer to [10]. Our references for -algebras are [11, 12].

For a -algebra , an element is positive () if and . denotes the set of 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 sesquilinear, 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 .

Example 1. (see [14]). If is a countable set of Hilbert -modules, then one can define their direct sum . On the -module of all sequences , such that the series is norm-convergent in the -algebra , we define the inner product byfor .
Hence, is a Hilbert -module.
The direct sum of a countable number of copies of a Hilbert -module is denoted by .
Let and be two Hilbert -modules. A map is said to be adjointable if there exists a map such that for all and .
We also reserve the notation for the set of all adjointable operators from to and is abbreviated to .
Let be the set for all positive bounded linear invertible operators on with bounded inverse.
The following lemmas will be used to prove our main results.

Lemma 1 (see [15]). If is a -homomorphism between -algebras, then is increasing, that is, if , then .

Lemma 2 (see [15]). Let and be two Hilbert -modules and .(i)If is injective and has closed range, then the adjointable map is invertible and(ii)If is surjective, then the adjointable map is invertible and

Lemma 3 (see [16]). Let be a Hilbert -modules. If , then

Lemma 4 (see [17]). 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

2. Controlled -Operator Frames for

We begin this section with the following definition.

Definition 2. (see [6]). A family of adjointable operators on a Hilbert -module over a unital -algebra is said to be an operator frames for , if there exist two positive constants such thatThe numbers and are called lower and upper bound of the operator frames, respectively. If , the operator frame is -tight.
If , it is called a normalized tight operator frames or a Parseval operator frames.
If only upper inequality of (5) holds, then is called an operator Bessel sequence for .
If the sum in the middle of (5) is convergent in norm, the operator frame is called standard.

Definition 3. Let , and a family of adjointable operators on a Hilbert -module over a unital -algebra is said to be a -controlled - operator frames for if there exists two strictly nonzero elements and in such thatThe elements and are called lower and upper bound of the -controlled -operator frames, respectively.
If , the -controlled -operator frame is called -tight.
If , it is called a normalized tight -controlled -operator frames or a Parseval -controlled -operator frames.
If only upper inequality of (6) holds, then is called an -controlled -operator Bessel sequence for .

Example 2. Let be the unitary -algebra of all bounded complex-valued sequences and let , the set of all sequences converging to zero equipped with the -inner product:It is clear to see that is a Hilbert -module over .
Let and , and we define byLet , and we define two operators and on byWe haveTherefore, is a -controlled tight -operator frame for .

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

Proof. Let be a -controlled -operator frames for .
Then, there exist two positives constants such thatHence,Therefore, is a -controlled -operator frames for with bounds and .
Let be a -controlled -operator frame for .
The bounded linear operator is given bywhich is called the analysis operator for the -controlled -operator frame .
The adjoint operator is given bywhich is called the synthesis operator for the -controlled -operator frames .
When and commute with each other and commute with the operator for each , we define the -controlled frame operator: by .
From now, we assume that and commute with each other and commute with the operator for each .

Proposition 2. The -controlled frame operator is bounded, positive, self-adjoint, and invertible.

Proof. Let be a -controlled -operator frames; then,and hence,Then,It is clear that is positive and bounded operator.
We haveTherefore, . Since and commute with each other and commute with , we have self-adjoint.
From the definition of controlled -operator frame, we haveSo,where is the identity operator in . Thus, is invertible.

3. Some Characterizations of Controlled -Operator Frames

Theorem 1. Let such that converge in norm . Then, is a -controlled -operator frames if and only iffor every and strictly nonzero elements in .

Proof. Suppose that is a -controlled -operator frames; then, we haveHence,Conversely, assume that (21) holds. From (13), the -controlled frame operator is positive, self-adjoint, and invertible. Then, we have, for all ,Using (21) and (24), we obtainUsing (25) and Lemma 4, we conclude that is a -controlled -operator frames .
The following theorem shows that any -operator frame is a -controlled -operator frames for and vice versa.

Theorem 2. Let . The family is a -operator frame for if and only if is a -controlled -operator frames.

Proof. Let be a -controlled -operator frames with bounds A and B. Then,On the one hand, for all , we haveThen,On the other hand, for any , we haveThen,Therefore, is a -operator frames with bounds and .
For the converse, suppose that is a -operator frames with bounds M and N.
On the one hand, we have, for any ,Thus, for all , we haveOn the other hand, we haveTherefore,This gives that is a -controlled -operator frame with bounds and .

Proposition 3. Let be a -operator frames for with frame operator S and let . Suppose that commute with each other and commute with for all ; then, is a -controlled -operator frame for .

Proof. Let be an -operator frame with bounds and . Then, by (5), we haveHence,We haveUsing (36) and (38), we haveTherefore, from Theorem 1, we conclude that is a -controlled -operator frame with bounds and .

Theorem 3. Let and . Suppose that commute with each other and commute with for all . The family is a -controlled -operator Bessel sequence for with bound if and only if the operator given byis well defined, bounded, and .

Proof. Assume that is a -controlled -operator Bessel sequence for with bound . As a result of (36),We haveThen, the sum is convergent, and we haveHence,Thus, the operator is well defined, bounded, andFor the converse, suppose that the operator is well defined, bounded, and . For all , we havewhere .
Therefore,Hence,This give that is a -controlled -operator Bessel sequence for .

Theorem 4. Let be a -controlled -operator frames for with bounds A and B, with operator frame . Let be injective and has a closed range. Suppose that commute with and . Then, is a -controlled -operator frames for with frame operator with bounds and .

Proof. Let be a -controlled -operator frames for with bounds A and B; then,From Lemma 2, we haveHence,Sincewe haveUsing (49), (51), and (53), we haveTherefore, is a -controlled -operator frame for . Moreover, for every , we haveThis completes the proof.

Corollary 1. Let be a -controlled -operator frame for , with frame operator . Then, is a -controlled -operator frame for .

Proof. The proof is a result of (53) for .

Theorem 5. Let be a -controlled -operator frame for with bounds A and B. Let be surjective. Then, is a -controlled -operator frame for with bounds and .

Proof. From the definition of -controlled -operator frame, we haveUsing Lemma 2, we haveFrom (56) and (57), we haveHence, is a -controlled -operator frame for .
Under those conditions, a controlled - operator frame for with a -module over a unital -algebras is also a controlled -operator frame for with a -module over a unital -algebras . The following theorem answers these questions.

Theorem 6. Let and be two hilbert -modules and let : be a -homomorphism and be a map on such that for all . Suppose is a -controlled -operator frame for with frame operator and lower and upper bounds A and B, respectively. If is surjective such that for each and and , then is a -controlled -operator frame for with frame operator and lower and upper bounds and , respectively, and .

Proof. Since is surjective, then, for every , there exists such that . Using the definition of -controlled -operator frame, we haveBy Lemma 1, we haveFrom the definition of -homomorphism, we haveUsing the relation between and , we obtainSince , , and , we haveTherefore,This implies that is a -controlled -operator frame for with bounds and . Moreover, we havewhich completes the proof [18].

Data Availability

No data were used to support this study.

Conflicts of Interest

The authors declare that there are no conflicts of interest.