Journal of Function Spaces
Volume 2018 (2018), Article ID 2108580, 6 pages
https://doi.org/10.1155/2018/2108580

Research Article

A New Inequality for Frames in Hilbert Spaces

Zhong-Qi Xiang

College of Mathematics and Computer Science, Shangrao Normal University, Shangrao, Jiangxi 334001, China

Correspondence should be addressed to Zhong-Qi Xiang; lxsy20110927@163.com

Received 15 July 2018; Accepted 20 September 2018; Published 3 October 2018

Academic Editor: Dashan Fan

Copyright © 2018 Zhong-Qi Xiang. 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 obtain a new inequality for frames in Hilbert spaces associated with a scalar and a bounded linear operator induced by two Bessel sequences. It turns out that the corresponding results due to Balan et al. and Găvruţa can be deduced from our result.

1. Introduction

A frame for a Hilbert space firstly emerged in the work on nonharmonic Fourier series owing to Duffin and Schaeffer [1], which has made great contributions to various fields because of its nice properties; the reader can examine the papers [2–12] for background and details of frames.

Balan et al. in [13] showed us a surprising inequality when they further investigated the Parseval frame identity derived from their study on efficient algorithms for signal reconstruction, which was then extended to general frames and alternate dual frames by Găvruţa [14]. In this paper, we establish a new inequality for frames in Hilbert spaces, where a scalar and a bounded linear operator with respect to two Bessel sequences are involved, and it is shown that our result can lead to the corresponding results of Balan et al. and Găvruţa.

The notations , , and are reserved, respectively, for a complex Hilbert space, the identity operator on , and an index set which is finite or countable. The algebra of all bounded linear operators on is designated as .

One says that a family of vectors in is a frame, if there are two positive constants satisfying

The frame is said to be Parseval if . If satisfies the inequality to the right in (1), we call that is a Bessel sequence for .

For a given frame , the frame operator , a positive, self-adjoint, and invertible operator on , is defined by

from which we see that

where the involved frame is said to be the canonical dual of .

For any , denote . A positive, bounded linear, and self-adjoint operator induced by and the frame is given below

Suppose that and are two Bessel sequences for . An application of the Cauchy-Schwartz inequality can show that the operator

is well-defined and further . Particularly, if , then both and are frames for . In this case we say that is an alternate dual frame of , and the pair is called an alternate dual frame pair.

2. The Main Results

We need the following simple result on operators to present our main result.

Lemma 1. Suppose that and that . Then for each we have

Proof. A direct calculation gives

From this fact and taking into account that

we arrive at the relation stated in the lemma.

We can immediately get the following result obtained by Poria in [15], when putting in Lemma 1.

Corollary 2. Suppose that and that . Then for every we have

Theorem 3. Suppose that is a frame for , that and are two Bessel sequences for , and that the operator is defined by (5). Then for each and each , we have

Moreover, if is self-adjoint, then for any and any ,

Proof. We take and for any . Then and further

By Lemma 1 we have

Therefore,

It follows that

We now prove the inequality in (10). Again by Lemma 1,

for every . Hence,

from which we conclude that

The proof of (11) is similar to the proof of (10); we leave the details to the reader.

Corollary 4. Suppose that is a frame for with frame operator and that for any . Then for all , for any and any , we have

Proof. Setting for each , then . Taking

then and are both Bessel sequences for . For any we have

A similar discussion yields

We also have

Thus the result follows from Theorem 3.

Let be a Parseval frame for ; then . Thus for any ,

Similarly we have

This together with Corollary 4 leads to a result as follows.

Corollary 5. Suppose that is a Parseval frame for . Then for each , for any and any , we have

Corollary 6. Suppose that is an alternate dual frame pair for . Then for each , for any and any , we have

Proof. Since is an alternate dual frame of , . For any , let

On the one hand we have

On the other hand we have

By Theorem 3 the conclusion follows.

Remark 7. Theorems 2.2 and 3.2 in [14] and Proposition 4.1 in [13] can be obtained when taking , respectively, in Corollaries 4, 6, and 5.

As a matter of fact, we can establish a more general inequality for alternate dual frames than that shown in Corollary 6.

Theorem 8. Suppose that is an alternate dual frame pair for . Then for every bounded sequence , for all and all , we have

Proof. We define the operators and by

Then both series converge unconditionally and . Since , by Corollary 2 we obtain

for each . Hence

Therefore,

It follows that

This completes the proof.

Remark 9. If we take in Theorem 8, then we can obtain Theorem 3.3 in [14].

Data Availability

No data were used to support this study.

Conflicts of Interest

The author declares that he has no conflicts of interest.

Acknowledgments

This work is supported by the National Natural Science Foundation of China (Grant Nos. 11761057 and 11561057).

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