Decompositions of -Frames and Duals and Pseudoduals of -Frames in Hilbert Spaces

Abstract

Firstly, we study the representation of -frames in terms of linear combinations of simpler ones such as -orthonormal bases, -Riesz bases, and normalized tight -frames. Then, we study the dual and pseudodual of -frames, which are critical components in reconstructions. In particular, we characterize the dual -frames in a constructive way; that is, the formulae for dual -frames are given. We also give some -frame like representations for pseudodual -frame pairs. The operator parameterizations of -frames and decompositions of bounded operators are the key tools to prove our main results.

1. Introduction

A sequence of elements of a Hilbert space is called a frame for if there are constants so that The numbers and are called the lower (resp., upper) frame bounds. The frame is a tight frame if and a normalized tight frame if .

The concept of frame first appeared in the late 40s and early 50s (see [1–3]). The development and study of wavelet theory during the last decades also brought new ideas and attention to frames because of their close connections. There are many related references on this topic, see [4–8].

In [9], Sun raised the concept of -frame as follows, which generalized the concept of frame extensively. A sequence is called a -frame for with respect to , which is a sequence of closed subspaces of a Hilbert space , if there exist two positive constants and such that, for any , where is called the lower -frame bound and is called the upper -frame bound. The largest lower frame bound and the smallest upper frame bound are called the optimal lower -frame bound and the optimal upper -frame bound, respectively. We simply call a -frame for whenever the space sequence is clear. The tight -frame and normalized tight -frame are defined similarly. We call a -frame sequence, if it is a -frame for . We call a -Bessel sequence, if only the right inequality is satisfied. A -frame for is called an alternate dual -frame of , if for every , we have If is a -frame for , then the operator such that is called the -fame operator associated with . It is well-known that is a dual -frame of , which is called the canonical dual -frame associated with . In this paper, we use dual of -frames to denote any of the duals. Recently, -frames in Hilbert spaces have been studied intensively; for more details, see [10–16] and the references therein.

Frames and -frames have advantages of allowing decomposing and reconstructing elements in Hilbert spaces, in which the dual and pseudodual of frames (-frames) play important roles. Characterizing dual frames and general frame decompositions is an important problem in pure and applied fields. In [17–22], the authors study the dual frames and frame-like decompositions in Hilbert spaces. In particular, Li derived a general parametric and algebraic formula for all duals of a frame in [17] and introduced the pseudoframe decompositions in [18]. In this paper, motivated by these works on frames, we consider similar problems on -frames in Hilbert spaces and generalize the corresponding results on frames to -frames. Another interesting problem in frame theory is representing general -frames in terms of special and more simpler -frames such as -orthonormal bases, -Riesz bases, and normalized tight -frames. In [23], the authors study similar questions on frames in Hilbert spaces by applying the techniques of decomposing linear bounded operators. In this paper, we will study the decompositions of -frames in Hilbert spaces by using similar techniques combing with what we have obtained on the operator parameterizations for -frames in [24].

Throughout this paper, we use to denote the set of natural numbers and to denote the complex plane. All Hilbert spaces in this paper are assumed to be separable complex Hilbert spaces. This paper is organized as follows. In Section 2, we give some definitions and lemmas which are needed to understand the following sections. In Section 3, we consider the decomposition of -frames. In Section 4, the dual and pseudodual of -frames are considered.

2. Preliminary Definitions and Lemmas

In this section, we introduce some basic definitions and lemmas which are necessary for the following sections.

Definition 1. Let , .(i)If , then we say that is -complete.(ii)If is -complete and there are positive constants and such that, for any finite subset and , ,  then we say that is a -Riesz basis for with respect to .(iii)We say is a -orthonormal basis for with respect to if it satisfies the following:

Remark 2. It is obvious that any -frame is -complete and any -orthonormal basis is a normalized tight -frame.

Definition 3. Suppose that , for any . If, for any , we have , then we call and a pair of pseudodual -frames for . In particular, if is a -frame for , we call a pseudodual -frame of .

Lemma 4 (see [25]). Let be a Hilbert space. Then,(1)for every invertible operator , there exists a unique decomposition , where is a unitary operator and is a positive operator.(2)for every positive operator with , , where is a unitary operator.

Lemma 5 (see [26]). Given Hilbert space and a sequence of closed subspaces of a Hilbert space , then there exists a -orthonormal basis for with respect to if and only if .

Lemma 6 (see [24]). Let be a -orthonormal basis for with respect to . Then, the sequence is a -Bessel sequence for if and only if there is a unique bounded operator such that for all .

Remark 7. Given the -orthonormal basis , the operator in Lemma 6 is called the -preframe operator associated with .

Lemma 8 (see [24]). Suppose that is a -orthonormal basis for , is a -Bessel sequence for , and and are the -preframe operator and -frame operator associated with , respectively. Then(1) is a -frame if and only if is onto;(2) is a normalized tight -frame if and only if is a coisometry;(3) is a -Riesz basis if and only if is invertible;(4) is a -orthonormal basis if and only if is unitary.

Lemma 9 (see [23]). Let be onto; then can be written as a linear combination of two unitary operators if and only if is invertible.

3. Decompositions of -Frames

In this section, we do some research on the decompositions of -frames in Hilbert spaces by using similar techniques in [23] combing with what we have established on the operator parameterizations for -frames in [24].

Theorem 10. Suppose that is a -Bessel sequence for . Let be the -preframe operator associated with . Then, for any , there exist three -orthonormal bases    such that for any .

Proof. Since we have assumed that all Hilbert spaces are separable, the -orthonormal bases for with respect to exist by Lemma 5. Let be a -orthonormal basis for with respect to . Since is a -Bessel sequence for , there exists a bounded operator such that for any by Lemma 6. Define an operator by , where is the identity operator on . Since is invertible. By Lemma 4, there exist a unitary and a positive operator such that . Since , where is a unitary operator by Lemma 4. So Hence, Denote , , and for any . Then, it is easy to see that    are -orthonormal bases for , since and are unitary operators. So for any .

Since a -frame is of course a -Bessel sequence, the following corollary is obvious.

Corollary 11. Every -frame can be represented as a multiple of sum of three -orthonormal bases.

Theorem 12. A -frame for can be written as a linear combination of two -orthonormal bases for if and only if is a -Riesz basis for .

Proof. Suppose that and are -orthonormal bases for such that for any . By Lemma 8, there exist surjective operator and unitary operator such that and for any . So, , . Hence, , . It implies that , since . So is invertible by Lemma 9. Hence, is a -Riesz basis for .
Since is a -Riesz basis for , there exist a -orthonormal basis and an invertible operator such that for any by Lemma 8. There exist two unitary operators and in and constants such that by Lemma 9. So for any . Since and are -orthonormal bases for by Lemmas 8 and 9, the result follows.

Theorem 13. Every -frame for is a multiple of two normalized tight -frames for .

Proof. Suppose that is a -frame for and is a -orthonormal basis for . Then, there exists a surjective operator such that for any by Lemma 8. Let . Then, and is also surjective. Suppose that is the polar decomposition of , where is a coisometry and is a positive operator in . Since , then with being a unitary operator. So . It follows that . So Since and are coisometries, and are normalized tight -frames for by Lemma 8. This finishes the proof.

Theorem 14. Every -frame for is a multiple of the sum of a -orthonormal basis for and a -Riesz basis for .

Proof. Suppose that is a -frame for and is a -orthonormal basis for . Let be the -preframe operator associated with ; then for any . Define operator by ; then So is invertible. Let be the polar decomposition of . Then, is a unitary operator and is a positive operator by Lemma 4. Since , . So by Lemma 4, where is a unitary operator. So . It implies that where . Since is invertible. Hence, is invertible. So Since is a -orthonormal basis for and is a -Riesz basis for by Lemma 8, is a multiple of a sum of a -orthonormal basis and a -Riesz basis for .

4. Dual and Pseudodual -Frames

In this section, we consider the characterizations of dual and pseudodual -frames. The algebraic formula about the dual of -frames for a given -frame will be given and some properties on dual and pseudodual -frames will be established.

Theorem 15. Suppose that is a -frame for and is a -orthonormal basis for . Suppose that the -preframe operator associated with is ; that is, for any . Then, is a dual -frame of if and only if for any , where is a bounded left inverse of .

Proof. Suppose that is a dual -frame of . Let be the -preframe operator of . Then, for any and is bounded. Since, for any , we have , It implies that . Hence, . It follows that is a bounded left inverse of .
Since , is bounded surjective operator in . Hence, is a -frame for by Lemma 8. Since is a dual -frame for .

Lemma 16. Suppose that is a -frame for , whose -preframe operator is . Then, is a linear bounded left inverse of if and only if where is the -frame operator associated with , , and   is the identity operator in .

Proof. Suppose is a linear bounded left inverse of . Let . Then Suppose . Then Hence, is a linear bounded left inverse of .

Theorem 17. Suppose is a -frame for , is its -preframe operator, and is its -frame operator. Let be a -orthonormal basis for . Then, is a dual -frame of if and only if there exists a bounded operator such that

Proof. Suppose that is a dual -frame of . Then, by Theorem 15, we know that for any , where is a linear bounded left inverse of . By Lemma 16, for some linear bounded operator . Hence, for any , we have Suppose that there exists a linear bounded operator such that . Then, So is a -Bessel sequence for and the -preframe operator associated with is Since is a linear bounded left inverse of by Lemma 16, . Therefore, is a dual -frame of by Theorem 15.

Theorem 18. Suppose that is a -frame for . If is a pseudodual -frame of , then has lower -frame bound.

Proof. Since is a pseudodual -frame of , for any . In particular, ; that is, . Since where is the upper -frame bound of , hence, So has lower -frame bound.

Corollary 19. Suppose that is a -frame for and is a pseudodual -frame of ; then is -complete.

Proof. Since has lower -frame bound by Theorem 18, there exists a constant such that If , for all , then ; it follows that . So is -complete.

Theorem 20. Suppose that and are a pair of pseudodual -frames for . Then, for any , if and only if , where the series converge in norm of .

Proof. It is obvious that we only need to prove one direction; the other direction is identical. Now, suppose that . Since and are a pair of pseudodual -frames for , we have . It is obvious that is a weakly continuous function on and for each . Since the closed unit ball of is weakly compact, for any , there exists such that, for any and any , we have . So whenever , we have Hence, .

Corollary 21. Suppose that and are a pair of pseudo -frames, . If is convergent, then

Proof. Since and are a pair of pseudo -frames, for any , we have So . It follows that by Theorem 20.

Theorem 22. Suppose that is a -orthonormal basis for and and are -preframe operators associated with -frames and , respectively. Then, and are -biorthogonal if and only if , where is the identity operator in .

Proof. Since and are -preframe operators associated with -frames and , respectively, and for any . So, for any and any , , we have If and are -biorthogonal, then So It implies that .
Conversely, if , then So and are -biorthogonal.

Conflict of Interests

The author declares that there is no conflict of interests regarding the publication of this paper.

Acknowledgment

This work was partially supported by the Fundamental Research Funds for the Central Universities (JBK140925).