On Schauder Frames in Conjugate Banach Spaces

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 ℓ_∞ 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], 𝐺-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 [11–14]. 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. Φ-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 ℓ_∞ has weak*-Schauder frame. Finally, a sufficient condition for the existence of a Schauder frame sequence has been given.

2. Preliminaries

Throughout this paper, 𝐸 will denote a Banach space, 𝐻 will denote a Hilbert space, let 𝐸^∗ the dual space of 𝐸, [𝑥_𝑛] be the closed linear span of {𝑥_𝑛} in the norm topology of 𝐸, and let 𝜋 be the canonical mapping of 𝐸 into 𝐸^∗∗. A series ∑^∞_𝑛=1𝑓_𝑖 in a conjugate Banach space 𝐸^∗ is called weak-convergent to 𝑓 if it converges in 𝜎(𝐸^∗,𝐸^∗∗)-topology. In this case, we write 𝑓=^𝑤∑^∞_𝑛=1𝑓_𝑛. A series ∑^∞_𝑛=1𝑓_𝑖 in a conjugate Banach space 𝐸^∗ is called weak*-convergent to 𝑓 if it converges in 𝜎(𝐸^∗,𝐸)-topology. In this case, we write 𝑓=^𝑤∗∑^∞_𝑛=1𝑓_𝑛.

Definition 1. A sequence {𝑥_𝑛}_𝑛∈ℕ⊂𝐻 is called a frame for 𝐻 if there exist 𝐴, 𝐵 with 0<𝐴≤𝐵<∞ such that 𝐴‖𝑥‖²≤^∞_𝑛=1𝑥,𝑥_𝑛²≤𝐵‖𝑥‖²,𝑥∈𝐻. (1)
The positive constants 𝐴 and 𝐵, respectively, are called lower and upper frame bounds of the frame {𝑥_𝑛}_𝑛∈ℕ. The inequality (1) is called the frame inequality.

Definition 2. Let 𝐸 be a Banach space. A sequence {(𝑥_𝑛,𝑓_𝑛)}_𝑛∈ℕ({𝑥_𝑛}⊂𝐸,{𝑓_𝑛}⊂𝐸^∗) is called a Schauder frame for 𝐸 if 𝑥=^∞_𝑛=1𝑓_𝑛(𝑥)𝑥_𝑛,𝑥∈𝐸. (2)

Definition 3. Let 𝐸 be a Banach space. A sequence {(𝑥_𝑛,𝑓_𝑛)}_𝑛∈ℕ({𝑥_𝑛}⊂𝐸,{𝑓_𝑛}⊂𝐸^∗) is called a Schauder frame sequence for 𝐸 if {(𝑥_𝑛,𝑓_𝑛)}_𝑛∈ℕ is a Schauder frame for [𝑥_𝑛].

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

Definition 4. Let 𝐸 be a Banach space and let 𝐸^∗ be its conjugate space. Let (𝐸^∗)_𝑑 be a Banach space of scalar-valued sequences associated with 𝐸^∗, indexed by ℕ. Let {𝑥_𝑛}⊂𝐸 and 𝑇∶(𝐸^∗)_𝑑→𝐸^∗ be given. The pair ({𝑥_𝑛},𝑇) is called a retro Banach frame for 𝐸^∗ with respect to (𝐸^∗)_𝑑 if(i){𝑓(𝑥_𝑛)}∈(𝐸^∗)_𝑑, 𝑓∈𝐸^∗, (ii)there exist positive constants 𝐴 and 𝐵 with 0<𝐴≤𝐵<∞ such that 𝐴𝑓_𝐸∗≤𝑓𝑥_{𝑛(𝐸∗)𝑑}≤𝐵𝑓_𝐸∗,𝑓∈𝐸^∗, (3)(iii)𝑇 is a bounded linear operator such that 𝑇𝑓𝑥_𝑛=𝑓,𝑓∈𝐸^∗. (4)
The positive constants 𝐴 and 𝐵, respectively, are called lower and upper frame bounds of the retro Banach frame ({𝑥_𝑛},𝑇). The operator 𝑇∶(𝐸^∗)_𝑑→𝐸^∗ is called the reconstruction operator (or the preframe operator). The inequality (3) is called the retro frame inequality. A retro Banach frame ({𝑥_𝑛},𝑇)({𝑥_𝑛}⊂𝐸,𝑇∶(𝐸^∗)_𝑑→𝐸^∗) for 𝐸 with respect to (𝐸^∗)_𝑑 with bounds 𝐴, 𝐵 is said to be tight, if it is possible to choose 𝐴=𝐵, normalized tight, if 𝐴=𝐵=1, and exact, if there exists no reconstruction operator 𝑇₀ such that ({𝑥_𝑛}_𝑛≠𝑗,𝑇₀)  (𝑗∈ℕ) is a retro Banach frame 𝐸^∗.
Finally, we give the following results which will be used in the subsequent results.

Theorem 5 (see [25]). If {𝑔_𝑛}⊂(ℓ_∞)^∗ and lim_𝑛→∞𝑔_𝑛(𝑥)=0, for all 𝑥∈ℓ_∞, then lim_𝑛→∞Φ(𝑔_𝑛)=0 for all Φ∈𝐸^∗∗.

Theorem 6 (see [24]). Let ({𝑥_𝑛},𝑇)({𝑥_𝑛}⊂𝐸,𝑇∶(𝐸^∗)_𝑑→𝐸^∗) be a retro Banach frame for 𝐸^∗ with respect to (𝐸^∗)_𝑑. Then, ({𝑥_𝑛},𝑇) is exact if and only if 𝑥_𝑛∉[𝑥_𝑖]_𝑖≠𝑛, for all 𝑛∈ℕ.

In view of Theorem 6, one may observe that if ({𝑥_𝑛},𝑇) is an exact retro Banach frame for 𝐸^∗, then there exists a sequence {𝑔_𝑛} in 𝐸^∗, called an admissible sequence to the retro Banach frame ({𝑥_𝑛},𝑇), such that 𝑔_𝑖(𝑥_𝑗)=𝛿_𝑖𝑗, for all 𝑖, 𝑗∈ℕ.

3. Main Results

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

Definition 7. Let 𝐸 be a Banach space, let {𝑓_𝑛} be a sequence in 𝐸^∗ and let {Φ_𝑛} be a sequence in 𝐸^∗∗, and let 𝑀 be a subset of 𝐸^∗. Then, {(𝑓_𝑛,Φ_𝑛)}_𝑛∈ℕ is said to be(a)weak-Schauder frame for 𝐸^∗ with respect to 𝑀 if 𝑓=^𝑤∞_𝑛=1Φ_𝑛𝑓𝑓_𝑛,𝑓∈𝑀, (5)(b)weak*-Schauder frame for 𝐸^∗ with respect to 𝑀 if 𝑓=^𝑤∗∞_𝑛=1Φ_𝑛𝑓𝑓_𝑛,𝑓∈𝑀. (6)
In particular, if 𝑀=𝐸^∗, then {(𝑓_𝑛,Φ_𝑛)}_𝑛∈ℕ is simply called a weak-Schauder frame and 𝑤𝑒𝑎𝑘^∗-Schauder frame for 𝐸^∗, respectively. Further, one may observe that if {(𝑓_𝑛,Φ_𝑛)}_𝑛∈ℕ is a Schauder frame for 𝐸^∗, then it is also a weak*-Schauder frame (weak-Schauder frame) for 𝐸^∗.

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

Example 8. ℓ^∗_∞ has no weak*-Schauder frame. Assume on the contrary that ℓ^∗_∞ has a weak*-Schauder frame {(𝑓_𝑛,Φ_𝑛)}_𝑛∈ℕ  ({𝑓_𝑛}⊂ℓ^∗_∞,{Φ_𝑛}⊂ℓ_∞^∗∗). Then, we have 𝑓(𝑥)=^∞_𝑖=1Φ_𝑖𝑓𝑓_𝑖(𝑥),𝑥∈𝐸,𝑓∈ℓ^∗_∞, (7) that is, 𝑓(𝑥)−^𝑛_𝑖=1Φ_𝑖𝑓𝑓_𝑖(𝑥)→0,as𝑛→∞. (8) So, by Theorem 5, we have lim_𝑛→∞Φ𝑓−^𝑛_𝑖=1Φ_𝑛𝑓𝑓_𝑖=0,Φ∈ℓ_∞^∗∗. (9) This gives ℓ^∗_∞ is weak-separable. This is not possible. Hence, ℓ^∗_∞ has no weak*-Schauder frame.

In view of Example 8 we have the following problem.

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

In this direction, we have the following result.

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

Proof. Let {(𝑥_𝑛,𝑓_𝑛)}_𝑛∈ℕ be a Schauder frame for 𝐸; then 𝑥=^∞_𝑖=𝑙𝑓_𝑖(𝑥)𝑥_𝑖,𝑥∈𝐸. (10) Therefore, for each 𝑓∈𝐸^∗, we have 𝑓(𝑥)=𝑓^∞_𝑖=1𝑓_𝑛(𝑥)𝑥_𝑛=^∞_𝑛=1𝜋𝑥_𝑛𝑓𝑓_𝑛(𝑥),𝑥∈𝐸. (11)

Corollary 10. ℓ_∞ has weak*-Schauder frame.

Proof . It follows from Proposition 9.

Note. ℓ_∞ does not have a Schauder frame.

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

Theorem 11. Let 𝐸 be a Banach space. Let {(𝑓_𝑛,Φ_𝑛)}_𝑛∈ℕ be a weak*-Schauder frame for 𝐸^∗ such that each Φ_𝑛 is weak*-continuous. Then, there exists a sequence {𝑥_𝑛}⊂𝐸 such that {(𝑥_𝑛,𝑓_𝑛)}_𝑛∈ℕ is a Schauder frame for 𝐸.

Proof. For each 𝑓∈𝐸^∗, we have ^∞_𝑛=1Φ_𝑛𝑓𝑓=_𝑛^𝑤∗𝑓. (12) Since each Φ_𝑛 is weak*-continuous, there exists a sequence {𝑥_𝑛}⊂𝐸 such that Φ_𝑛=𝜋(𝑥_𝑛),  𝑛∈ℕ. So, we have 𝑓(𝑥)=^∞_𝑛=1𝜋𝑥_𝑛𝑓𝑓_𝑛(𝑥)=𝑓^∞_𝑛=1𝑓_𝑛(𝑥)𝑥_𝑛,𝑥∈𝐸,𝑓∈𝐸^∗. (13) This gives 𝑥=^∞_𝑛=1𝑓_𝑛(𝑥)𝑥_𝑛,𝑥∈𝐸. (14)

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

Theorem 12. Let 𝐸 be a Banach space. Let {𝑓_𝑛} be a sequence in 𝐸^∗ and let {Φ_𝑛} be a sequence in 𝐸^∗∗. Then, {(𝑓_𝑛,Φ_𝑛)}_𝑛∈ℕ is a weak*-Schauder frame for 𝐸^∗ if and only if {(𝑓_𝑛,Φ_𝑛)}_𝑛∈ℕ is a weak-Schauder frame for 𝐸^∗ with respect to 𝜋(𝐸).

Proof. {(𝑓_𝑛,Φ_𝑛)}_𝑛∈ℕ is a weak*-Schauder frame for 𝐸^∗ if and only if (𝜋(𝑥))𝑓=^∞_𝑛=1Φ_𝑛𝑓𝜋(𝑥)𝑓_𝑛,𝑓∈𝐸^∗,𝜋(𝑥)∈𝜋(𝐸) (15) if and only if {(𝑓_𝑛,Φ_𝑛)}_𝑛∈ℕ is a weak-Schauder frame for 𝐸^∗ with respect to 𝜋(𝐸).

Remark 13. Let 𝐸 be a reflexive Banach space. Let {𝑓_𝑛} be a sequence in 𝐸^∗ and {Φ_𝑛} let be a sequence in 𝐸^∗∗. Then, by Theorem 12, {(𝑓_𝑛,Φ_𝑛)}_𝑛∈ℕ is a weak*-Schauder frame for 𝐸^∗ if and only if {(𝑓_𝑛,Φ_𝑛)}_𝑛∈ℕ is a weak-Schauder frame for 𝐸^∗.

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

Theorem 14. Let 𝐸 be a Banach space and let ({𝑥_𝑛},𝑇) be an exact retro Banach frame for 𝐸^∗ with admissible sequence {𝑓_𝑛}⊂𝐸^∗. Then, {(𝑥_𝑛,𝑓_𝑛)}_𝑛∈ℕ is a Schauder-frame sequence in 𝐸.

Proof. For each 𝑛∈ℕ, define 𝑆_𝑛∶𝐸→𝐸 by 𝑆_𝑛(𝑥)=^𝑛_𝑖=1𝑓_𝑖(𝑥)𝑥_𝑖,𝑥∈𝐸. (16) Let 𝑆^∗_𝑛 be the adjoint operator to 𝑆_𝑛. Then 𝑆^∗_𝑛𝑔(𝑥)=𝑔^𝑛_𝑖=1𝑓_𝑖(𝑥)𝑥_𝑖=^𝑛_𝑖=1𝑔𝑥_𝑖𝑓_𝑖(𝑥),𝑥∈𝐸,𝑔∈𝐸^∗,𝑛=1,2,3…. (17) This gives 𝑆^∗_𝑛𝑔=^𝑛_𝑖=1𝑔𝑥_𝑖𝑓_𝑖,𝑔∈𝐸^∗,𝑛=1,2,3,…. (18) Thus, for every finite linear combination 𝑔=∑^𝑚_𝑖=1𝛽_𝑖𝑓_𝑖, we have 𝑆^∗_𝑛𝑔=^𝑛_𝑖=1𝑔𝑥_𝑖𝑓_𝑖=^𝑚_𝑖=1𝛽_𝑖𝑓_𝑖=𝑔,𝑛=𝑚,𝑚+1,…. (19) Let 𝑓∈[𝑓_𝑛] and 𝜀>0 be given. Then, there exists a finite linear combination 𝑔=∑^𝑚𝜀_𝑗=1𝛽_𝑗^(𝜀)𝑓_𝑗 such that ‖𝑓−𝑔‖<𝜀/(𝜈+1), where 𝜈=sup_1≤𝑛<∞‖𝑠_𝑛‖<∞. Also, 𝑠^∗_𝑛𝑓−𝑓≤𝑆^∗_𝑛𝑓−𝑆^∗_𝑛𝑔+𝑆^∗_𝑛𝑔+<−𝑔𝑔−𝑓𝜈𝜀+𝜀𝜈+1𝜈+1=𝜀,𝑛=𝑚_𝜀,𝑚_𝜀+1,…. (20)

Acknowledgment

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