#### 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.