#### Abstract

Frame theory has a great revolution for recent years. This theory has been extended from Hilbert spaces to Hilbert -modules. In this paper, we define and study the new concept of controlled continuous ---frames for Hilbert -modules and we establish some properties.

#### 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 frame and Gabor frame [3, 4]. Frames have been used in signal processing, image processing, data compression, and sampling theory. The concept of a generalization of frame to a family indexed by some locally compact space endowed with a Radon measure was proposed by Kaiser [5] and independently by Ali et al. [6]. These frames are known as continuous frames. Gabardo and Han in [7] called them frames associated with measurable spaces, Askari-Hemmat et al. in [8] called them generalized frames, and in mathematical physics, they are known as coherent states.

In 2012, Gavruta [9] introduced the notion of -frame in Hilbert space to study the atomic systems with respect to a bounded linear operator . Moreover, the following two recent papers [10, 11] are closely linked to Gravuta’s paper and represent a generalization of it by considering also unbounded operators.

Controlled frames in Hilbert spaces have been introduced by Balazs et al. [12] to improve the numerical efficiency of iterative algorithms for inverting the frame operator.

Frank and Larson [13] were the first one that extend frames to Hilbert -modules.

Controlled frames in Hilbert -modules was introduced by Kouchi and Rahimi [14], where the authors showed that they share many useful properties with their corresponding notions in a Hilbert space. Finally, we note that controlled --frames in Hilbet spaces have been introduced by Hua and Huang [15]. The theory of continuous frames has been generalized in Hilbert -modules. For more details, see [16–22].

In this paper, we introduce the notion of a controlled continuous ---frame for Hilbert -modules.

In the following, we briefly recall the definitions and basic properties of -algebras and Hilbert -modules. Our references for -algebras are [23, 24]. For a -algebra , if is positive, we write and denotes the set of all positive elements of .

*Definition 1. *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 it 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 .

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 2 [25]. *Let and be two Hilbert -modules and . Then, the following statements are equivalent:
*(i)

*is surjective*(ii)

*is bounded below with respect to the norm; i.e., there is such that ,*(iii)

*is bounded below with respect to the inner product; i.e., there is such that*

Lemma 3. *Let and be two Hilbert -modules and . Then, the following statements are equivalent:
*(i)*The operator is bounded and -linear*(ii)*There exist such that for all *

Lemma 4 [26]. *Let and be two Hilbert -modules and . Then,
*(i)

*if is injective and has a closed range, then the adjointable map is invertible and*

*(ii)*

*if is surjective, then the adjointable map is invertible and*

For the following theorem, denote the range of the operator .

Theorem 5 [27]. *Let be a Hilbert -module over a -algebra and let be two operators for . If is closed, then the following statements are equivalent:
*(i)

*for some*(i)

*There exists such that*

#### 2. Controlled Continuous ---Frames for Hilbert -Modules

Let be a Banach space, be a measure space, and be a measurable function. Integral of the 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. Since every -algebra and Hilbert -module are a Banach space, thus we can use this integral and its properties.

Let and be two Hilbert -modules, is a family of subspaces of , and is the collection of all adjointable -linear maps from into . We define

For any and , if the -valued inner product is defined by , the norm is defined by . The is a Hilbert -module.

Let be a -algebra; is defined by

is a Hilbert -module (Hilbert -module) with pointwise operations and the inner product defined by

Let be the set of all positive bounded linear invertible operators on with bounded inverse.

*Definition 6 [16]. *Let be a family in for all , and . We say that the family *is a**-*controlled continuous -frame for Hilbert *-*module with respect to if it is a continuous *-*Bessel family and there is a pair of constants such that, for any ,

and are called the -controlled continuous -frame bounds.

If , then we call a -controlled continuous -frame for with respect to .

*Definition 7. **Let*. The family is called a *-*controlled continuous *-g-*frame for Hilbert *-*module with respect to if is continuous *-g-*Bessel sequence and if there exist two strictly nonzero elements and in such that

and are called the -controlled continuous -g-frame bounds.

If , then we call a -controlled continuous -g-frames for with respect to .

*Definition 8. *Let be a Hilbert *-*module over a unital *-*algebra, and . A family of adjointable operators is said to be a *-*controlled continuous *-**-**-*frame for Hilbert *-*module with respect to if there exist two strictly nonzero elements and in such that

The elements and are called continuous -controlled ---frame bounds.

If we call this continuous -controlled ---frame a continuous -controlled tight --frame, and if it is called a continuous -controlled Parseval ---frame. If only the right-hand inequality of (10) is satisfied, we call a continuous -controlled Bessel --frame with Bessel bound .

*Example 1. *Let , and .

It is known that is a -algebra and is a Hilbert -module.

Let and be two operators, respectively, defined as follows: where and are two real numbers strictly positive.

It is clear that .

Indeed, for each , one has

Let endowed with the Lebesgue measure. It is clear that there is a measure space. Moreover, for , we define the operator by

is linear, bounded, and selfadjoint.

In addition, for , we have which show that the family is a continuous -controlled Bessel sequence for with as bounded.

Let by

because is a projection. Furthermore, we have for every .

Then, is a continuous -controlled --frame for .

*Remark 9. *Every *-*controlled continuous *-**-*frame for is a *-*controlled continuous *-**-**-*frame for .

Indeed, if is a -controlled continuous --frame for Hilbert -module with respect to , then there exist two strictly nonzero elements and in such that

But

So,

Then,

Hence, is a controlled continuous ---frame for Hilbert -module with respect to with bounds and .

*Remark 10. *Let be a surjective operator. Every -controlled continuous ---frame for is a -controlled continuous --frame for .

*Proof. *Let be a surjective operator; then, there exists such that
Hence,
Since is a -controlled continuous ---frame, then there exist two strictly nonzero elements and in such that
Therefore,
This gives that is a -controlled continuous --frame with bounds and , respectively.

Proposition 11. *Let and . Suppose that and commute with each other and commute with the operators for each . A family is a -controlled continuous ---frame Bessel family for with respect to with bounds if and only if the operator
is well defined and bounded with .*

*Proof. *Assume that is a -controlled continuous Bessel ---frame family for with respect to with bounds . As a result of (2.3),
Then, the sum is convergent and we have
Hence,
Thus, the operator is well defined and bounded and
For the converse, suppose that the operator is well defined and bounded and For all , we have
where .

Therefore,
Hence,
This gives that is a -controlled continuous ---frame Bessel family for with respect to .

Let be a -controlled continuous Bessel ---frame for Hilbert -module over with respect to with bounds and .

We define the operator by such that

The bounded linear operator is called the synthesis operator of .

The operator given by is called the analysis operator for .

Indeed, we have for all and which shows that is the adjoint of . If and commute between them and commute with the operators for each , we define the -controlled continuous ---frame operator by

As consequence, one has the following proposition.

Proposition 12. *The operator is positive, selfadjoint, and bounded and
*

If is surjective, then is invertible.

*Proof. *By definition, we have Then, is a selfadjoint.

Clearly, is positive.

By definition of a -controlled continuous ---frame, we have
So
This gives
If we take supremum on all , where , we have
If is surjective, then, using Lemma 4, we have which is invertible.

Lemma 13. *Let be a -controlled continuous Bessel ---frame family for Hilbert -module with respect to . Then, for any such that and , the family is a -controlled continuous Bessel ---frame for Hilbert -module .*

*Proof. *Assume that is a -controlled continuous Bessel ---frame family for Hilbert -module with respect to with bound . Then,
So,
Hence,
The results hold.

Lemma 14. *Let and . Let be a -controlled continuous Bessel ---frame for Hilbert -module with respect to . is a-controlled continuous ---frame if and only if there exists such that
*

*Proof. *The family is a -controlled continuous ---frame if and only if
if and only if
if
and the family is a continuous -controlled Bessel --frame sequence; then,
which completes the proof.

Theorem 15. *Let and . Suppose that commute with and . If is a -controlled continuous --frame for Hilbert -module with respect to , then is a -controlled continuous ---frame for Hilbert -module with respect to with frame operator .*

*Proof. *Let be a -controlled continuous --frame for Hilbert -module with respect to ; then,
Hence,
Therefore,
Thus,
This concludes that is a -controlled continuous ---frame for Hilbert -module with respect to .

Proposition 16. *Let and . Suppose that and commute with each other and commute with S. Then, is a -controlled continuous ---frame for with respect to if and only if is a -controlled continuous ---frame for Hilbert -module with respect to .*

*Proof. *For all , we have
where
Hence,
For any , we have
Hence, is a continuous -controlled ---frame for with bounds and with respect to if and only if
The results hold.

Proposition 17. *Let and . Then, is a -controlled continuous ---frame for Hilbert -module with respect to if and only if is a continuous -controlled ---frame for with respect to .*

*Proof. *The proof is similar as the proof of Lemma 2.12.

Proposition 18. *Let and . Suppose that , , and . Then, is a -controlled continuous ---frame for with respect to if and only if is a continuous ---frame for with respect to .*

*Proof. *Assume that is a continuous ---frame for with bounds and with respect to . Then,
Since are bounded positive operators, there exist constants , such that
From
we have
Then,
Therefore,
So
Hence, is a continuous -controlled ---frame for with respect to .

Conversely, assume that is a -controlled continuous ---frame for with respect to with bounds and . On the one hand, we have for any ,
So,
On the other hand, we have