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Abstract and Applied Analysis
Volume 2011 (2011), Article ID 361595, 20 pages
http://dx.doi.org/10.1155/2011/361595
Research Article

Continuous g-Frame in Hilbert -Modules

1Department of Mathematics, Science and Research Branch, Islamic Azad University, Kerman 7635131167, Iran
2Department of Mathematics, Faculty of Mathematics and Computer, Shahid Bahonar University of Kerman, Kerman 7616914111, Iran

Received 18 August 2010; Revised 23 November 2010; Accepted 19 January 2011

Academic Editor: H. B. Thompson

Copyright © 2011 Mehdi Rashidi Kouchi and Akbar Nazari. This is an open access article distributed under the Creative Commons Attribution License, which permits unrestricted use, distribution, and reproduction in any medium, provided the original work is properly cited.

Abstract

We give a generalization of g-frame in Hilbert -modules that was introduced by Khosravies then investigated some properties of it by Xiao and Zeng. This generalization is a natural generalization of continuous and discrete g-frames and frame in Hilbert space too. We characterize continuous g-frame g-Riesz in Hilbert -modules and give some equality and inequality of these frames.

1. Introduction

Frames for Hilbert spaces were first introduced in 1952 by Duffin and Schaeffer [1] for study of nonharmonic Fourier series. They were reintroduced and development in 1986 by Daubechies et al. [2] and popularized from then on. The theory of frames plays an important role in signal processing because of their importance to quantization [3], importance to additive noise [4], as well their numerical stability of reconstruction [4] and greater freedom to capture signal characteristics [5, 6]. See also [79]. Frames have been used in sampling theory [10, 11], to oversampled perfect reconstruction filter banks [12], system modelling [13], neural networks [14] and quantum measurements [15]. New applications in image processing [16], robust transmission over the Internet and wireless [1719], coding and communication [20, 21] were given. For basic results on frames, see [4, 12, 22, 23].

let be a Hilbert space, and a set which is finite or countable. A system is called a frame for if there exist the constants such that for all . The constants and are called frame bounds. If we call this frame a tight frame and if it is called a Parseval frame.

In [24] Sun introduced a generalization of frames and showed that this includes more other cases of generalizations of frame concept and proved that many basic properties can be derived within this more general context.

Let and be two Hilbert spaces, and is a sequence of subspaces of , where is a subset of Z. is the collection of all bounded linear operators from into . We call a sequence a generalized frame, or simply a -frame, for with respect to if there are two positive constants and such that for all . The constants and are called -frame bounds. If we call this -frame a tight g-frame, and if it is called a Parseval g-frame.

On the other hand, the concept of frames especially the g-frames was introduced in Hilbert -modules, and some of their properties were investigated in [2527]. Frank and Larson [25] defined the standard frames in Hilbert -modules in 1998 and got a series of result for standard frames in finitely or countably generated Hilbert -modules over unital -algebras. As for Hilbert -module, it is a generalization of Hilbert spaces in that it allows the inner product to take values in a -algebra rather than the field of complex numbers. There are many differences between Hilbert -modules and Hilbert spaces. For example, we know that any closed subspace in a Hilbert space has an orthogonal complement, but it is not true for Hilbert -module. And we cannot get the analogue of the Riesz representation theorem of continuous functionals in Hilbert -modules generally. Thus it is more difficult to make a discussion of the theory of Hilbert -modules than that of Hilbert spaces in general. We refer the readers to papers [28, 29] for more details on Hilbert -modules. In [27, 30], the authors made a discussion of some properties of g-frame in Hilbert -module in some aspects.

The concept of a generalization of frames to a family indexed by some locally compact space endowed with a Radon measure was proposed by Kaiser [23] and independently by Ali at al. [31]. These frames are known as continuous frames. Let be a Hilbert space, and let be a measure space. A continuous frame in indexed by is a family such that (a)for any , the function defined by is measurable;(b)there is a pair of constants such that, for any , If and is the counting measure, the continuous frame is a frame.

The paper is organized as follows. In Sections 2 and 3 we recall the basic definitions and some notations about continuous g-frames in Hilbert -module; we also give some basic properties of g-frames which we will use in the later sections. In Section 4, we give some characterization for continuous g-frames in Hilbert -modules. In Section 5, we extend some important equalities and inequalities of frame in Hilbert spaces to continuous frames and continuous g-frames in Hilbert -modules.

2. Preliminaries

In the following we review some definitions and basic properties of Hilbert -modules and g-frames in Hilbert -module; we first introduce the definition of Hilbert -modules.

Definition 2.1. Let be a -algebra with involution . An inner product -module (or pre-Hilbert -module) is a complex linear space which is a left -module with map which satisfies the following properties:(1) for all ;(2) for all and ;(3) for all ;(4) for all and if and only if .For , we define a norm on by . Let is complete with this norm, it is called a Hilbert -module over or a Hilbert -module.

An element of a -algebra is positive if and spectrum of is a subset of positive real number. We write to mean that is positive. It is easy to see that for every , hence we define .

Frank and Larson [25] defined the standard frames in Hilbert -modules. If be a Hilbert -module, and a set which is finite or countable. A system is called a frame for if there exist the constants such that for all . The constants and are called frame bounds.

A. Khosravi and B. Khosravi [27] defined g-frame in Hilbert -module. Let and be two Hilbert -module, and is a sequence of subspaces of , where is a subset of and is the collection of all adjointable -linear maps from into that is, for all and . We call a sequence a generalized frame, or simply a g-frame, for Hilbert -module with respect to if there are two positive constants and such that for all . The constants and are called g-frame bounds. If we call this g-frame a tight g-frame, and if it is called a Parseval g-frame.

Let be a measure space, let and be two Hilbert -modules, is a sequence of subspaces of , and is the collection of all adjointable -linear maps from into .

Definition 2.2. We call a net a continuous generalized frame, or simply a continuous g-frame, for Hilbert -module with respect to if(a)for any , the function defined by is measurable;(b)there is a pair of constants such that, for any , The constants and are called continuous g-frame bounds. If we call this continuous g-frame a continuous tight g-frame, and if it is called a continuous Parseval g-frame. If only the right-hand inequality of (2.3) is satisfied, we call the continuous g-Bessel for with respect to with Bessel bound .

If and is the counting measure, the continuous g-frame for with respect to is a g-frame for with respect to .

Let be a Banach space, a measure space, and function a measurable function. Integral of the Banach-valued function has defined Bochner and others. Most properties of this integral are similar to those of the integral of real-valued functions for example triangle inequality. The reader is referred to [32, 33] for more details. Because every -algebra and Hilbert -module is a Banach space thus we can use this integral and its properties.

Example 2.3. Let be a Hilbert -module on -algebra , and let be a frame for . Let be the functional induced by Then is a g-frame for Hilbert -module with respect to .

Example 2.4. If is admissible, that is, and, for and then is a continuous frame for with respect to equipped with the measure and, for all , where is the continuous wavelet transform defined by For details, see [12, Proposition and Corollary ].

3. Continuous g-Frame Operator and Dual Continuous g-Frame on Hilbert -Algebra

Let be a continuous g-frame for with respect to . Define the continuous g-frame operator on by

Lemma 3.1 (see [33]). Let be a measure space, and are two Banach spaces, be a bounded linear operator and measurable function; then

Proposition 3.2. The frame operator is a bounded, positive, selfadjoint, and invertible.

Proof. First we show, is a selfadjoint operator. By Lemma 3.1 and property (3) of Definition 2.1 for any we have It is clear that we have Now we show that is a bounded operator Inequality (2.3) and equality (3.4) mean that or in the notation from operator theory ; thus is a positive operator. Furthermore, , and consequently ; this shows that is invertible.

Proposition 3.3. Let be a continuous g-frame for with respect to with continuous g-frame operator with bounds and . Then defined by is a continuous g-frame for with respect to with continuous g-frame operator with bounds and . That is called continuous canonical dual g-frame of .

Proof. Let be the continuous g-frame operator associated with that is . Then for , Hence .
Since is a continuous g-frame for , then . On other hand since and are selfadjoint and commutative with and , , and hence .

Remark 3.4. We have . In other words and are dual continuous g-frame with respect to each other.

4. Some Characterizations of Continuous g-Frames in Hilbert -Module

In this section, we will characterize the equivalencies of continuous g-frame in Hilbert -module from several aspects. As for Theorem 4.2, we show that the continuous g-frame is equivalent to which the middle of (2.3) is norm bounded. As for Theorems 4.3 and 4.6, the characterization of g-frame is equivalent to the characterization of bounded operator .

Lemma 4.1 (see [34]). Let be a -algebra, and two Hilbert -modules, and . Then the following statements are equivalent: (1) is surjective;(2) is bounded below with respect to norm, that is, there is such that for all ;(3) is bounded below with respect to the inner product, that is, there is such that .

Theorem 4.2. Let for any . Then be a continuous g-frame for with respect to if and only if there exist constants such that for any

Proof. Let is a continuous g-frame for with respect to . Then inequality (4.1) is an immediate result of -algebra theory.
If inequality (4.1) holds, then by Proposition 3.2, , hence for any . Now by use of Lemma 4.1, there are constants such that which implies that is a continuous g-frame for with respect to .

We define

For any and , if the -valued inner product is defined by , the norm is defined by , then is a Hilbert -module (see [28]).

Let be a continuous g-frame for with respect to , we define synthesis operator by; for all . So analysis operator is defined for map by for any .

Theorem 4.3. A net is a continuous g-frame for with respect to if and only if synthesis operator is well defined and surjective.

Proof. Let be a continuous g-frame for with respect to ; then operator is well defined and because
For any , by that is invertible, there exist such that . Since is a continuous g-frame for with respect to , so and , which implies that is surjective.
Now let be a well-defined operator. Then for any we have It follow that .
On the other hand, since is surjective, by Lemma 4.1, is bounded below, so is invertible. Then for any , we have , so . It is easy to check that Hence .

Corollary 4.4. A net is a continuous g-Bessel net for with respect to if and only if synthesis operator is well defined and .

Definition 4.5. A continuous g-frame in Hilbert -module with respect to is said to be a continuous g-Riesz basis if it satisfies that (1) for any ;(2)if , then every summand is equal to zero, where and is a measurable subset of .

Theorem 4.6. A net is a continuous g-Riesz for with respect to if and only if synthesis operator is homeomorphism.

Proof. We firstly suppose that is a continuous g-Bessel net for with respect to . By Theorem 4.3 and that it is g-frame, is surjective. If for some , according to the definition of continuous g-Riesz basis we have for any , and , so for any , namely . Hence is injective.
Now we let the synthesis operator be homeomorphism. By Theorem 4.3   is a continuous g-frame for with respect to . It is obviouse that for any . Since is injective, so if , then , so . Therefore is a continuous g-Riesz for with respect to .

Theorem 4.7. Let be a continuous g-frame for with respect to , with g-frame bounds . Let for any . Then the following are equivalent:(1) is a continuous g-frame for with respect to ;(2)there exists a constant , such that for any , one has

Proof. First we let be a continuous g-frame for with respect to with bounds . Then for any , we have Similarly we can obtain Let ; then inequality (4.7) holds.
Next we suppose that inequality (4.7) holds. For any , we have
Also we can obtain

Next we will introduce a bounded operator about two g-Bessel sequences in Hilbert -module. The idea is derived from the operator which was considered for fusion frames by Găvruţa in [35]. In this paper, we will use the operator to characterize the g-frames of Hilbert -module further. Let and be two g-Bessel sequences for with respect to , with Bessel bounds , respectively. Then there exists a well-defined operator As a matter of fact, for any , we have It is easy to know that and .

Theorem 4.8. Let be a continuous g-frame for with respect to , with g-frame bounds . Suppose that is a continuous g-Bessel net for with respect to . If is surjective, then is a continuous g-frame for with respect to .
On the contrary, if is a continuous g-Riesz basis for with respect to , then is surjective.

Proof. Suppose that is a continuous g-frame for with respect to . By Theorem 4.3, we can define the synthesis operator of (4.12). It is easy to check that the adjoint operator of is analysis operator as follows: for any .
On the other hand, since is a continuous g-Bessel net for with respect to , by Corollary 4.4 we also can define the corresponding operator .
Hence we have for any , namely, . Since is surjective, then for any , there exists such that , and , it follows that is surjective. By Theorem 4.3 we know that is a continuous g-frame for with respect to .
On the contrary, suppose that is a continuous g-Riesz basis and is a continuous g-frame for with respect to . By Theorem 4.6, is homeomorphous, so is . By Theorem 4.3   is surjective, therefore is surjective.

Theorem 4.9. Let be a continuous g-Riesz basis, is a continuous g-Bessel net for with respect to . Then a continuous g-Riesz basis for with respect to if and only if is invertible.

Proof. We first suppose that is invertible. Since is a g-Riesz basis, is a g-Bessel sequence for with respect to , by Theorem 4.3 and Corollary 4.4 we can define the operators mentioned before and is homeomorphous, hence is also invertible. From the proof of Theorem 4.7 we know that . Since is invertible, so is . By Theorem 4.6 we have that is a g-Riesz basis for with respect to .
Now we let and be two g-Riesz basis for with respect to . By Theorem 4.6 both are invertible, so is invertible too.

5. Some Equalities for Continuous g-frames in Hilbert -Modules

Some equalities for frames involving the real parts of some complex numbers have been established in [36]. These equalities generalized in [30] for g-frames in Hilbert -modules. In this section, we generalize the equalities to a more general form which generalized before equalities and we deduce some equalities for g-frames in Hilbert -modules to alternate dual g-frame.

In [37], the authors verified a longstanding conjecture of the signal processing community: a signal can be reconstructed without information about the phase. While working on efficient algorithms for signal reconstruction, the authors of [38] established the remarkable Parseval frame equality given below.

Theorem 5.1. If is a Parseval frame for Hilbert space , then for any and , one has

Theorem 5.1 was generalized to alternate dual frames [36]. If is a frame, then frame is called alternate dual frame of if for any , .

Theorem 5.2. If is a frame for Hilbert space and is an alternate dual frame of , then for any and , one has

Recently, Zhu and Wu in [39] generalized equality (5.2) to a more general form which does not involve the real parts of the complex numbers.

Theorem 5.3. If is a frame for Hilbert space and is an alternate dual frame of , then for any and , one has

Now, we extended this equality to continuous g-frames and g-frames in Hilbert -modules and Hilbert spaces. Let be a Hilbert -module. If is a continuous g-frame for with respect to , then continuous g-frame is called alternate dual continuous g-frame of if for any , .

Lemma 5.4 (see [30]). Let be a Hilbert -module. If are two bounded -linear operators in and , then one has

Now, we present main theorem of this section. In following, some result of this theorem for the discrete case will be present.

Theorem 5.5. Let be a continuous g-frame, for Hilbert -module with respect to and continuous g-frame is alternate dual continuous g-frame of , then for any measurable subset and , one has

Proof. For any measurable subset , let the operator be defined for any by .
Then it is easy to prove that the operator is well defined and the integral it is finite. By definition alternate dual continuous g-frame . Thus, by Lemma 5.4 we have Hence the theorem holds. The proof is completed.

Corollary 5.6. Let be a discrete g-frame, for Hilbert -module with respect to , and discrete g-frame is alternate dual discrete g-frame of , then for any subset and , one has

Corollary 5.7. Let be a g-frame, for Hilbert space with respect to and g-frame is alternate dual g-frame of , then for any measurable subset and , one has

The following results generalize the results in [30] in the case of continuous g-frames.

Lemma 5.8 (see [30]). Let be a Hilbert -module. If is a bounded, selfadjoint linear operator and satisfy , for all , then .

Lemma 5.9 (see [30]). Let be a Hilbert -module. If are two bounded, selfadjoint -linear operators in and , then one has

Theorem 5.10. Let be a continuous g-frame, for Hilbert -module with respect to and let be the canonical dual continuous g-frame of , then for any measurable subset and , one has

Proof. Since is an invertible, positive operator on , and , then . Let . By Lemma 5.9, we obtain Replacing by , then one has On the other hand, we have
Associating with (5.12) the proof is finished.

Corollary 5.11. Let be a continuous frame for Hilbert -module with canonical dual frame , then for any measurable subset and , one has

Proof. For any , if we let in Theorem 5.10, then we get the conclusion.

Theorem 5.12. Let be a continuous Parseval g-frame, for Hilbert -module with respect to , then for any measurable subset and , one has

Proof. Since is a continuous Parseval g-frame in Hilbert -module with respect to , then for any , we have So Hence for any , we have . Let . Since is bounded, selfadjoint, then , so is also bounded, selfadjoint. By Lemma 5.8, we have , namely, , so . From (5.16), then we have that: for any measurable subset and , Combining (5.16) and Theorem 5.10, we get the result.

Corollary 5.13. Let be a continuous Parseval frame for Hilbert -module , then for any measurable subset and , one has

Corollary 5.14. Let be a continuous -tight g-frame, for Hilbert -module with respect to , then for any measurable subset and , one has

Proof. Since is a continuous -tight g-frame, then is a continuous g-Parseval frame, by Theorem 5.12 we know that the conclusion holds.

Corollary 5.15. Let be a continuous -tight frame for Hilbert -module then for any measurable subset and , one has

Corollary 5.16. Let be a continuous -tight g-frame, for Hilbert -module with respect to , then for any measurable subset and , one has

Proof. Since for any , by Corollary 5.14, we get

Acknowledgment

The authors would like to give a special thanks to the referee(s) and the editor(s) for their valuable comments and suggestions which improved the presentation of the paper.

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