Some Generalizations of Frames in Hilbert Modules

Abstract

Frames play significant role in various areas of science and engineering. In this paper, we introduce the concept of frames for the set of all adjointable operators from to and their generalizations. Moreover, we obtain some new results for generalized frames in Hilbert modules.

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].

Traditionally, frames have been used in signal processing, image processing, data compression, and sampling theory. A discreet frame is a countable family of elements in a separable Hilbert space which allows for a stable, not necessarily unique, decomposition of an arbitrary element into an expansion of the frame elements.

Many generalizations of the concept of frame have been defined in Hilbert spaces and Hilbert -modules [4–9].

This paper generalizes the papers’ G-frames as special frames [10], Frames and Operator Frames for [11], and Generalized Frames for [12], in framework of the Hilbert -modules.

Let be a finite or countable index subset of . In this section, we briefly recall the definitions and basic properties of -algebra, Hilbert -modules, frames in Hilbert -modules, and their generalizations. For information about frames in Hilbert spaces, we refer to [13]. Our references for -algebras are [14, 15]. For a -algebra , if is positive, we write , and denotes the closed cone of positive elements in .

Definition 1. (see [14]). If is a Banach algebra, an involution is a map of into itself such that, for all and in and all scalar , the following conditions hold:(1)(2)(3)

Definition 2. (see [14]). A -algebra is a Banach algebra with involution such thatfor every in .

Examples 1. (1), the algebra of bounded operators on a Hilbert space , is a -algebra, where, for each operator , is the adjoint of (2), the algebra of continuous functions on a compact space , is an abelian -algebra, where (3), the algebra of continuous functions on a locally compact space that vanishes at infinity, is an abelian -algebra, where

Definition 3. (see [16]). 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 and positive definite and respects the module action. In other words,(1), for all and , if and only if (2), for all and (3), 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 .

Examples 2. Let be a locally compact Hausdorff space and a Hilbert space, and the Banach space of all continuous -valued functions vanishing at infinity is a Hilbert -module over the -algebra with inner product and module operation , for all and . 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 . Then, 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, and 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 .
is a Hilbert -module with the inner product , .

Lemma 1. (see [17]). Letandbe two Hilbert-modules and.(i)Ifis injective andhas closed range, then the adjointable mapis invertible and(ii)Ifis surjective, then the adjointable mapis invertible and

Definition 4. (see [18]). We call a sequence a g-frame in Hilbert -module with respect to if there exist two positive constants , , such that, for all ,where the numbers and are called lower and upper bounds of the g-frame, respectively. If , the g-frame is -tight. If , it is called a g-Parseval frame. If the sum in the middle of (5) is convergent in norm, the g-frame is called standard.

Definition 5. (see [17]). Let be a Hilbert -module over a unital -algebra. A family of elements of is a -frame for , if there exist strictly nonzero elements , in , such that, for all ,where the elements and are called lower and upper bounds of the -frame, respectively. If , the -frame is -tight. If , it is called a normalized tight -frame or a Parseval -frame. If the sum in the middle of (6) is convergent in norm, the -frame is called standard.

Definition 6. (see [19]). We call a sequence a -g-frame in Hilbert -module over a unital -algebra with respect to if there exist strictly nonzero elements , in , such that, for all ,where the elements and are called lower and upper bounds of the -g-frame, respectively. If , the -g-frame is -tight. If , it is called a -g-Parseval frame. If the sum in the middle of (7) is convergent in norm, the -g-frame is called standard.

Definition 7. (see [20]). Let and , for all ; then, is said to be a -g-frame for with respect to if there exist two constants such thatThe numbers and are called -g-frame bounds. Particularly, ifThe -g-frame is -tight.

Definition 8. (see [21]). Let . A family of elements of is a -K-frame for if there exist strictly nonzero elements and in , such that, for all ,where the elements and are called lower and upper bound of the --frame, respectively.

Definition 9. (see [22]). Let . We call a sequence a -K-g-frame in Hilbert -module with respect to if there exist strictly nonzero elements and in such thatThe numbers and are called lower and upper bounds of the -K-g-frame, respectively. IfThe -K-g-frame is -tight.

2. Some Results for Generalized Frames in Hilbert -Modules

We begin this section with the following hTeorem.

Theorem 1. Letbe a Hilbert-module over a commutative-algebra. A familyof elements ofis a-frame forif and only if there exist strictly positive elementsandin, such that, for all,

Proof. Let be a -frame for ; then, there exist strictly nonzero elements and in , such that, for all ,By commutativity, we haveWe pose and ; then, and are strictly positive elements in . Hence,Conversely, let and be strictly positive elements in , such that, for all ,where and are strictly positive elements in ; then, there exist and in such that and .
So,By commutativity, we haveTherefore, is an -frame for .

Corollary 1. A sequenceis a-g-frame in Hilbert-moduleover a commutative-algebra with respect toif and only if there exist strictly positive elements,in, such that for all,

Corollary 2. Let. A familyof elements ofis a-K-frame forif and only if there exist strictly positive elementsandin, such that, for all,

Corollary 3. Let. A sequenceis a-K-g-frame in Hilbert-moduleover a commutative-algebra, with respect toif and only if there exist strictly positive elementsandin, such that, for all,For a sequence of Hilbert -module , the space is a Hilbert -module with the inner product

Proposition 1. Let.(1)The sequenceis a g-frame forwith respect toif and only if the sequenceis a g-frame forwith respect to, with(2)The sequenceis a-g-frame forwith respect toif and only if the sequenceis a-g-frame forwith respect to, with(3)For, the sequenceis a-g-frame forwith respect toif and only if the sequenceis a-g-frame forwith respect to, with(4)For, the sequenceis a--g-frame forwith respect toif and only if the sequenceis a--g-frame forwith respect to, with

3. Frame for

We begin this section with the following definition.

Definition 10. A sequence is said to be a frame for if there exist such thatwhere the series converges in the strong operator topology.

Example 3. For , let such that, for all is injective and has a closed range.
Then, for all , we haveSo,Hence,Thus,for all .
This shows that is a frame for .

Example 4. Let such that, for all is injective and has a closed range and .
Then, for all , we haveSo,Hence,Thus,for all .
This shows that is a frame for .

Theorem 2. A sequenceis a frame forif and only if it is a g-frame forwith respect to.

Proof. Let be a g-frame for with respect to .
Then, there exist two positive constants and , such thatSo,Hence,Thus,i.e., is a frame for .
Conversely, assume that is a frame for .
Then, there exist two positive constants and , such thatSo,Let and such that ; then,i.e.,and thus, is a g-frame for with respect to .

Corollary 4. A sequenceis a tight frame forif and only if it is a tight g-frame forwith respect to.

Corollary 5. A sequenceis a Bessel sequence forif and only if it is a g-Bessel sequence forwith respect to.

4. Generalized Frame for

We begin this section with the following definition.

Definition 11. A sequence is said to be a generalized frame for if there exist strictly positive elements and in such thatwhere the series converges in the strong operator topology.
In the following, is a commutative -algebra.

Theorem 3. A sequenceis a generalized frame forif and only if it is a-g-frame forwith respect to.

Proof. Let be a -g-frame for with respect to .
Then, there exist strictly positive elements and in , such thatSo,Hence,Thus,i.e., is a generalized frame for .
Conversely, assume that is a generalized frame for .
Then, there exist strictly positive elements and in , such thatSo,Let and such that ; then,i.e.,and thus, is a -g-frame for with respect to .

Corollary 6. A sequenceis a generalized tight frame forif and only if it is a tight-g-frame forwith respect to.

Corollary 7. A sequenceis a generalized Bessel sequence forif and only if it is a-g-Bessel sequence forwith respect to.

5. K-Frame for

We begin this section with the following definition.

Definition 12. Let . A sequence is said to be a -frame for if there exist such thatwhere the series converges in the strong operator topology.

Example 5. For , let .
Then, for , we haveSo,Hence,This shows that is a Parseval -frame for .
Similar to Remark 1 in [22], we have the following.

Remark 1. (1)Every frame for is a -frame, for any : (2)If is a surjective operator, then every -frame for is a frame for

Theorem 4. Letandbe a frame for. Then,is a-frame for.

Proof. Let be a frame for .
Then, there exist two positive constants and , such thatHence,Thus,Then, is a -frame for .

Theorem 5. Letandbe a-frame for. Then,is a-frame for.

Proof. Let be a -frame for .
Then, there exist two positive constants and , such thatHence,Thus,Then, is a -frame for .

Corollary 8. Letandbe a-frame for. Then,is a-frame for.

Proof. It follows from the previous theorem.

Theorem 6. A sequenceis a-frame forif and only if it is a-g-frame forwith respect to.

Proof. Let be a -g-frame for with respect to .
Then, there exist two positive constants and , such thatSo,Hence,Thus,i.e., is a -frame for .
Conversely, assume that is a -frame for .
Then, there exist two positive constants and , such thatSo,Let and such that ; then,i.e.,and thus, is a -g-frame for with respect to .

Corollary 9. A sequenceis a tight-frame forif and only if it is a tight-g-frame forwith respect to.

6. Generalized K-Frame for

We begin this section with the following definition.

Definition 13. Let . A sequence is said to be a generalized -frame for if there exist strictly positive elements and in such thatwhere the series converges in the strong operator topology.
In the following, is a commutative -algebra.

Theorem 7. A sequenceis a generalized-frame forif and only if it is a--g-frame forwith respect to.

Proof. Letbe an--g-frame forwith respect to.

Then, there exist strictly positive elements and in , such that

So,

Hence,

Thus,i.e., is a generalized -frame for .

Conversely, assume that is a generalized -frame for .

Then, there exist strictly positive elements and in , such that

So,

Let and such that ; then,i.e.,and thus, is a --g-frame for with respect to .

Corollary 10. A sequenceis a generalized tight-frame forif and only if it is a tight--g-frame forwith respect to.

Data Availability

No data were used to support this study.

Disclosure

This manuscript is presented as preprint in arXiv (https://arxiv.org/pdf/1805.11655v2.pdf). Also, the manuscript has been submitted as a preprint in the following link: https://www.researchgate.net/publication/325464080.

Conflicts of Interest

There are no conflicts of interest.