#### Abstract

The concept of canonical dual -Bessel sequences was recently introduced, a deep study of which is helpful in further developing and enriching the duality theory of -frames. In this paper we pay attention to investigating the structure of the canonical dual -Bessel sequence of a Parseval -frame and some derived properties. We present the exact form of the canonical dual -Bessel sequence of a Parseval -frame, and a necessary and sufficient condition for a dual -Bessel sequence of a given Parseval -frame to be the canonical dual -Bessel sequence is investigated. We also give a necessary and sufficient condition for a Parseval -frame to have a unique dual -Bessel sequence and equivalently characterize the condition under which the canonical dual -Bessel sequence of a Parseval -frame admits a unique dual -Bessel sequence. Finally, we obtain a minimal norm property on expansion coefficients of elements in the range of resulting from the canonical dual -Bessel sequence of a Parseval -frame.

#### 1. Introduction

Throughout this paper, and are separable Hilbert spaces; is a finite or countable index set. We denote by the collection of all linear bounded operators from to , and is abbreviated as .

A sequence of elements in is a* frame* if there exist constants such that The frame is a* Parseval frame* if . If only the right-hand inequality holds, then is called a* Bessel sequence* with Bessel bound .

Associated with every Bessel sequence of there is a linear bounded operator, called the* analysis operator* of , defined by It is easy to check that the adjoint of , , is given by By composing and , we obtain the* frame operator *: Note that is a positive, self-adjoint operator, and it is invertible if and only if is a frame. Recall that a Bessel sequence in is a* dual frame* of if It is well-known that is a dual frame of , which is called the* canonical dual frame*.

Frames were formally defined by Duffin and Schaeffer [1] in the early 1950s, when they were used to study some deep problems on nonharmonic Fourier series. Owing to the redundancy and flexibility, today they have served as an important tool in various fields; see [2–10] for more information on frame theory and its applications. Atomic systems for subspaces were first introduced by Feichtinger and Werther in [11] based on examples arising in sampling theory. When working on atomic systems for operators, Găvruţa [12] put forward the concept of -frames for a given linear bounded operator , which allows atomic decomposition of elements from the range of and, in general, the range may not be closed. Moreover, it has been shown in [13–16] that in many ways -frames behave completely differently from frames, although a -frame is a generalization of a frame; see also [17, 18].

The classical canonical dual for a -frame is absent since the frame operator may not be invertible, which has greatly contributed to the fact that there are few results on the duals of a -frame. Recently, Guo in [15] proposed the concept of canonical dual -Bessel sequences from the operator-theoretic point of view, a deep study of which is helpful in further developing and enriching the duality theory of -frames. This paper is devoted to examining the structure of the canonical dual -Bessel sequence of a Parseval -frame and some derived properties. We present the exact form of the canonical dual -Bessel sequence of a Parseval -frame by means of the pseudo-inverse of and a necessary and sufficient condition for a dual -Bessel sequence of a given Parseval -frame to be the canonical dual -Bessel sequence. We also give a necessary and sufficient condition for a Parseval -frame to have a unique dual -Bessel sequence and equivalently characterize the condition for the canonical dual -Bessel sequence to admit a unique dual -Bessel sequence. We end the paper by showing that the canonical dual -Bessel sequence of a Parseval -frame gives rise to expansion coefficients of elements in the range of with minimal norm.

We need to collect some definitions and basic properties for operators.

*Definition 1. *Suppose . A sequence in is said to be a -frame, if there exist such that The constants and are called the lower and upper -frame bounds.

Suppose . A -frame of is said to be Parseval, if

*Definition 2. *Let be a sequence in . For , if implies for any , then we say that is -*linearly independent*.

The following results from operator theory will be used to prove our main results.

Lemma 3 (see [19]). *Suppose that has closed range, then there exists a unique operator , called the pseudo-inverse of , satisfying*

In the sequel, the notation is reserved for the pseudo-inverse of (if it exists).

Lemma 4 (see [20]). *Let and be two Hilbert spaces. Also let and . The following statements are equivalent. *(1)*(2)**There exists such that *(3)*There exists such that ** Moreover, if (1), (2), and (3) are valid, then there exists a unique operator such that*(a)*(b)**(c)*

Lemma 5. *Suppose that and is a Bessel sequence of with analysis operator . Then is a -frame of if and only if .*

*Proof. *It is an immediate consequence of Lemma 4; we omit the details.

#### 2. Main Results

Suppose and is a -frame of . From Theorem 3 in [12] we know that there always exists a Bessel sequence of such that which is called a* dual **-Bessel sequence* of (see Definition 2.5 in [15]). A direct calculation can show that a dual -Bessel sequence is necessarily a -frame.

In general, a -frame may admit more than one dual -Bessel sequence, as shown in the following example.

*Example 6. *Suppose that , , where Define as follows: Taking for , then for every . Therefore , showing that is a Parseval -frame of . Since it follows that is a dual -Bessel sequence of . Let and then it is easy to check that is Bessel sequence of . Now for any we have Hence is a dual -Bessel sequence of and is different from .

Guo in [15] proved that, among all dual -Bessel sequences of a given -frame, there is a unique dual -Bessel sequence whose analysis operator obtains the minimal norm, which is called the* canonical dual **-Bessel sequence*. Motivated by the idea of [21], in the following we characterize the exact structure of the canonical dual -Bessel sequence of a Parseval -frame under the condition that has closed range. We need the following two lemmas first. Since their proofs are similar to Theorems 2.7 and 2.8 in [21], respectively, we omit the details.

Lemma 7. *Suppose that has closed range and is a Parseval -frame of , then is a dual -Bessel sequence of .*

Lemma 8. *Suppose that has closed range and is a Parseval -frame of with analysis operator , then in is a dual -Bessel sequence of if and only if there exists such that and for every and every .*

By using Lemmas 7 and 8 we can obtain the following result which shows that the dual -Bessel sequence of Parseval -frame stated in Lemma 7 is exactly the canonical dual -Bessel sequence. For details of the proof, the reader can check the proof for Theorem 2.10 in [21], step by step.

Theorem 9. *Suppose that has closed range and is a Parseval -frame of with analysis operator , then is the canonical dual -Bessel sequence of .*

*Remark 10. * The canonical dual -Bessel sequence of the Parseval -frame , which will be denoted by later, is actually a Parseval frame on since for every , by Lemma 3.

The canonical dual -Bessel sequence of the Parseval -frame is precisely a Parseval -frame since Although the canonical dual -Bessel sequence of the Parseval -frame is not a Parseval -frame in general, it can naturally generate a new one in the form . Indeed, by Lemma 3 we have

We give a necessary and sufficient condition for a dual -Bessel sequence of a Parseval -frame to be the canonical dual -Bessel sequence.

Theorem 11. *Suppose that has closed range and is a Parseval -frame of with a dual -Bessel sequence . Then is the canonical dual -Bessel sequence of if and only if for any dual -Bessel sequence of , where and denote the analysis operators of and , respectively.*

*Proof. *Let us first assume that . If we denote by the analysis operator of , then a direct calculation can show that . From this fact and taking into account the fact that we obtain for any . Thus ; equivalently, .

Conversely, let for any dual -Bessel sequence of . Then and it follows that , implying that is the canonical dual -Bessel sequence of . This completes the proof.

A natural problem arises: under what condition will a Parseval -frame admit a unique dual -Bessel sequence? To this problem, we have the following.

Theorem 12. *Suppose that has closed range and is a Parseval -frame of with analysis operator . Then has a unique dual -Bessel sequence if and only if .*

*Proof. *Assume first that . Then is injective. Let and be two dual -Bessel sequences of ; then it is easy to check that for each and that Thus for any and and, consequently, for any .

For the opposite implication, assume contrarily that . Since is a Parseval -frame, it is easily seen that . Thus by Lemma 4, and has closed range as a consequence. Let be an invertible operator and . Taking and for each , then, for every , meaning that is a Bessel sequence of . Now let for every ; then it is easily seen that is a Bessel sequence of . Since is orthogonal to , for any . Therefore for any , which yields Since , there exists such that and, thus, , since a simple calculation gives . Hence is a dual -Bessel sequence of and is different from , a contradiction.

It is interesting that the -linear independence of a Parseval -frame can immediately lead to the -linear independence of and vice versa and that the uniqueness of the dual -Bessel sequence of implies the uniqueness of the dual -Bessel sequence of .

Theorem 13. *Suppose that has closed range and is a Parseval -frame of . Then the following results hold:** is -linearly independent if and only if is -linearly independent.** If admits a unique dual -Bessel sequence, then admits a unique dual -Bessel sequence.*

*Proof. * We first prove the necessity. Let be the analysis operator of ; then Therefore,Since is -linearly independent, it follows that for any and any , and for any as a consequence. Suppose now that for some , then Again by the -linear independence of we obtain for each .

For the sufficiency, let for . Then Thus for every , and the conclusion follows.

Since has a unique dual -Bessel sequence, by Theorem 12 we know that its analysis operator is surjective and, thus, is injective, which implies that is -linearly independent. Hence, by (1), is also -linearly independent, from which we conclude that has a unique dual -Bessel sequence.

We also present the following condition for the canonical dual -Bessel sequence of a Parseval -frame to have a unique dual -Bessel sequence.

Theorem 14. *Suppose that has closed range and is a Parseval -frame of . If the remainder of fails to be a new -frame whenever any one of its elements is deleted, then has a unique dual -Bessel sequence.*

*Proof. *To prove that admits a unique dual -Bessel sequence, it is sufficient to show that is -linearly independent, and it is enough to show that is -linearly independent by Theorem 13(1). Suppose on the contrary that there is with for some such that . Then . Denote by the analysis operator of ; then, by Lemma 5, . Thus for each there exists such that Clearly, is a Bessel sequence of and . Therefore, and, consequently, , where is the analysis operator of . Again by Lemma 5 it follows that is a -frame of , which leads to a contradiction.

At the end of the paper, we show that the canonical -dual Bessel sequences of Parseval -frames give rise to expansion coefficients of elements in with minimal norm.

Theorem 15. *Suppose that has closed range and is a Parseval -frame of . Then for any satisfying , we have *

*Proof. *It is easy to check that for every . Therefore,

#### Data Availability

No data were used to support this study.

#### Conflicts of Interest

The author declares that he has no conflicts of interest.

#### Acknowledgments

The research is supported by the National Natural Science Foundation of China (Nos. 11761057 and 11561057).