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Journal of Mathematics
Volume 2013 (2013), Article ID 318659, 4 pages
http://dx.doi.org/10.1155/2013/318659
Research Article

On Schauder Frames in Conjugate Banach Spaces

1Department of Mathematics, Kirori Mal College, University of Delhi, Delhi 110 007, India
2Department of Mathematics, Motilal Nehru College, University of Delhi, Delhi 110 021, India

Received 31 August 2012; Accepted 17 November 2012

Academic Editor: Ding-Xuan Zhou

Copyright © 2013 S. K. Kaushik et al. 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

Weak*-Schauder frame in conjugate Banach spaces has been introduced and studied. A sufficient condition for the existence of weak*-Schauder frame in the conjugate space of a separable Banach space has been given. It has been shown that has weak*-Schauder frame. Finally, a sufficient condition for the existence of a Schauder frame sequence has been given.

1. Introduction

Frames for Hilbert spaces were formally introduced by Duffin and Schaeffer [1]. Later, Daubechies et al. [2] found new applications to wavelets in which frames played an important role. Frames are main tools for use in signal and image processing, compression, sampling theory, optics, filter banks, signal detection, and so forth. In order to have more applications of frames, several notions generalizing the concept of frames have been introduced and studied, namely, pseudoframes [3], oblique frames [4], frames of subspaces (fusion frames) [5], -frames [6], orthogonal frames [7, 8], and so forth.

Feichtinger and Gröchenig [9] extended the notion of atomic decomposition to Banach spaces. Gröchenig [10] introduced a more general concept for Banach spaces called Banach frame. Banach frames and atomic decompositions were further studied in [1114]. Han and Larson [15] defined Schauder frame for a Banach space. In [16], Casazza, et al. gave various definitions of frames for Banach spaces including that of Schauder frame. In 2008, Casazza et al. [17] studied the coefficient quantization of Schauder frames in Banach spaces. Liu [18] gave the concepts of minimal and maximal associated bases with respect to Schauder frames and closely connected them with the duality theory. In [19], Liu and Zheng gave a characterization of Schauder frames which are near-Schauder bases. In fact, they generalized some results due to Holub [20]. Beanland et al. [21] proved that the upper and lower estimates theorems for finite dimensional decompositions of Banach spaces can be extended and modified to Schauder frames and gave a complete characterization of duality for Schauder frames. -Schauder frames were introduced and studied by Vashisht [22]. Recently, Liu [23] associated an operator with a Schauder frame and called it Hilbert-Schauder frame operator.

In the present paper, we introduce the concept of weak*-Schauder frame and weak-Schauder frame in conjugate Banach spaces. A sufficient condition for the existence of weak*-Schauder frame in a conjugate Banach space of a separable Banach space has been given. Also, an example of a conjugate space of a nonseparable Banach space which has no weak*-Schauder frame is given. Further, it has been shown that has weak*-Schauder frame. Finally, a sufficient condition for the existence of a Schauder frame sequence has been given.

2. Preliminaries

Throughout this paper, will denote a Banach space, will denote a Hilbert space, let the dual space of , be the closed linear span of in the norm topology of , and let be the canonical mapping of into . A series in a conjugate Banach space is called weak-convergent to if it converges in -topology. In this case, we write . A series in a conjugate Banach space is called weak*-convergent to if it converges in -topology. In this case, we write .

Definition 1. A sequence is called a frame for if there exist , with such that
The positive constants and , respectively, are called lower and upper frame bounds of the frame . The inequality (1) is called the frame inequality.

Definition 2. Let be a Banach space. A sequence is called a Schauder frame for if

Definition 3. Let be a Banach space. A sequence is called a Schauder frame sequence for if is a Schauder frame for .

Next, we give the definition of a retro Banach frame introduced in [24].

Definition 4. Let be a Banach space and let be its conjugate space. Let be a Banach space of scalar-valued sequences associated with , indexed by . Let and be given. The pair is called a retro Banach frame for with respect to if(i), , (ii)there exist positive constants and with such that (iii) is a bounded linear operator such that
The positive constants and , respectively, are called lower and upper frame bounds of the retro Banach frame . The operator is called the reconstruction operator (or the preframe operator). The inequality (3) is called the retro frame inequality. A retro Banach frame for with respect to with bounds , is said to be tight, if it is possible to choose , normalized tight, if , and exact, if there exists no reconstruction operator such that    is a retro Banach frame .
Finally, we give the following results which will be used in the subsequent results.

Theorem 5 (see [25]). If and , for all , then for all .

Theorem 6 (see [24]). Let be a retro Banach frame for with respect to . Then, is exact if and only if , for all .

In view of Theorem 6, one may observe that if is an exact retro Banach frame for , then there exists a sequence in , called an admissible sequence to the retro Banach frame , such that , for all , .

3. Main Results

We begin with the following definitions of weak*-Schauder frame and weak-Schauder frame in .

Definition 7. Let be a Banach space, let be a sequence in and let be a sequence in , and let be a subset of . Then, is said to be(a)weak-Schauder frame for with respect to if (b)weak*-Schauder frame for with respect to if
In particular, if , then is simply called a weak-Schauder frame and -Schauder frame for , respectively. Further, one may observe that if is a Schauder frame for , then it is also a weak*-Schauder frame (weak-Schauder frame) for .

Now, we give an example of a conjugate space of a nonseparable Banach space which has no weak*-Schauder frame.

Example 8. has no weak*-Schauder frame. Assume on the contrary that has a weak*-Schauder frame   . Then, we have that is, So, by Theorem 5, we have This gives is weak-separable. This is not possible. Hence, has no weak*-Schauder frame.

In view of Example 8 we have the following problem.

Problem 1. Does the conjugate space of a separable Banach space possess a weak*-Schauder frame?

In this direction, we have the following result.

Proposition 9. If a Banach space has a Schauder frame, then its conjugate Banach space has a weak*-Schauder frame.

Proof. Let be a Schauder frame for ; then Therefore, for each , we have

Corollary 10. has weak*-Schauder frame.

Proof . It follows from Proposition 9.

Note. does not have a Schauder frame.

Towards the converse of Proposition 9, we have the following result.

Theorem 11. Let be a Banach space. Let be a weak*-Schauder frame for such that each is weak*-continuous. Then, there exists a sequence such that is a Schauder frame for .

Proof. For each , we have Since each is weak*-continuous, there exists a sequence such that ,  . So, we have This gives

In the following result, we characterize weak*-Schauder frame in terms of weak-Schauder frame with respect to a subset.

Theorem 12. Let be a Banach space. Let be a sequence in and let be a sequence in . Then, is a weak*-Schauder frame for if and only if is a weak-Schauder frame for with respect to .

Proof. is a weak*-Schauder frame for if and only if if and only if is a weak-Schauder frame for with respect to .

Remark 13. Let be a reflexive Banach space. Let be a sequence in and let be a sequence in . Then, by Theorem 12, is a weak*-Schauder frame for if and only if is a weak-Schauder frame for .

Finally, we give a sufficient condition for the existence of a Schauder frame sequence in a Banach space.

Theorem 14. Let be a Banach space and let be an exact retro Banach frame for with admissible sequence . Then, is a Schauder-frame sequence in .

Proof. For each , define by Let be the adjoint operator to . Then This gives Thus, for every finite linear combination , we have Let and be given. Then, there exists a finite linear combination such that , where . Also,

Acknowledgment

The authors thank the referee(s) for providing valuable comments and useful suggestions for the improvement of the paper.

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