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

National Natural Science Foundation of China11761057115610571. Introduction

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

A sequence {fj}j∈J of elements in H is a frame if there exist constants 0<A≤B<∞ such that (1)Af2≤∑j∈Jf,fj2≤Bf2,∀f∈H.The frame {fj}j∈J is a Parseval frame if A=B=1. If only the right-hand inequality holds, then {fj}j∈J is called a Bessel sequence with Bessel bound B.

Associated with every Bessel sequence {xj}j∈J of H there is a linear bounded operator, called the analysis operator of {xj}j∈J, defined by (2)UX:H→l2J,UXf=f,xjj∈J.It is easy to check that the adjoint of UX, UX∗, is given by (3)UX∗:l2J→H,UX∗cjj∈J=∑j∈Jcjxj.By composing UX∗ and UX, we obtain the frame operator SX:H→H: (4)SXf=UX∗UXf=∑j∈Jf,xjxj.Note that SX is a positive, self-adjoint operator, and it is invertible if and only if {xj}j∈J is a frame. Recall that a Bessel sequence {yj}j∈J in H is a dual frame of {xj}j∈J if (5)f=∑j∈Jf,yjxj,∀f∈H.It is well-known that {SX-1xj}j∈J is a dual frame of {xj}j∈J, 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 K-frames for a given linear bounded operator K, which allows atomic decomposition of elements from the range of K and, in general, the range may not be closed. Moreover, it has been shown in [13–16] that in many ways K-frames behave completely differently from frames, although a K-frame is a generalization of a frame; see also [17, 18].

The classical canonical dual for a K-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 K-frame. Recently, Guo in [15] proposed the concept of canonical dual K-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 K-frames. This paper is devoted to examining the structure of the canonical dual K-Bessel sequence of a Parseval K-frame and some derived properties. We present the exact form of the canonical dual K-Bessel sequence of a Parseval K-frame by means of the pseudo-inverse of K and a necessary and sufficient condition for a dual K-Bessel sequence of a given Parseval K-frame to be the canonical dual K-Bessel sequence. We also give a necessary and sufficient condition for a Parseval K-frame to have a unique dual K-Bessel sequence and equivalently characterize the condition for the canonical dual K-Bessel sequence to admit a unique dual K∗-Bessel sequence. We end the paper by showing that the canonical dual K-Bessel sequence of a Parseval K-frame gives rise to expansion coefficients of elements in the range of K with minimal norm.

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

Definition 1.

Suppose K∈B(H). A sequence {fj}j∈J in H is said to be a K-frame, if there exist 0<C≤D<∞ such that (6)CK∗f2≤∑j∈Jf,fj2≤Df2,∀f∈H.The constants C and D are called the lower and upper K-frame bounds.

Suppose K∈B(H). A K-frame {fj}j∈J of H is said to be Parseval, if (7)K∗f2=∑j∈Jf,fj2,∀f∈H.

Definition 2.

Let {fj}j∈J be a sequence in H. For {cj}j∈J∈l2(J), if ∑j∈Jcjfj=0 implies cj=0 for any j∈J, then we say that {fj}j∈J is l2(J)-linearly independent.

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

Lemma 3 (see [<xref ref-type="bibr" rid="B19">19</xref>]).

Suppose that Λ∈B(H,K) has closed range, then there exists a unique operator Λ†∈B(K,H), called the pseudo-inverse of Λ, satisfying(8)ΛΛ†Λ=Λ,Λ†ΛΛ†=Λ†,ΛΛ†∗=ΛΛ†,Λ†Λ∗=Λ†Λ,KerΛ†=RangeΛ⊥,RangeΛ†=KerΛ⊥.

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

Lemma 4 (see [<xref ref-type="bibr" rid="B20">20</xref>]).

Let H1 and H2 be two Hilbert spaces. Also let S∈B(H1,H) and T∈B(H2,H). The following statements are equivalent.

Range(S)⊂Range(T)

There exists λ>0 such that SS∗≤λTT∗

There exists θ∈B(H1,H2) such that S=Tθ

Moreover, if (1), (2), and (3) are valid, then there exists a unique operator θ such that

θ2=infμ:SS∗≤μTT∗

KerS=Kerθ

Range(θ)⊂Range¯(T∗)

Lemma 5.

Suppose that K∈B(H) and {fj}j∈J is a Bessel sequence of H with analysis operator UF. Then {fj}j∈J is a K-frame of H if and only if Range(K)⊂Range(UF∗).

Proof.

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

2. Main Results

Suppose K∈B(H) and {fj}j∈J is a K-frame of H. From Theorem 3 in [12] we know that there always exists a Bessel sequence {gj}j∈J of H such that (9)Kf=∑j∈Jf,gjfj,∀f∈H,which is called a dual K-Bessel sequence of {fj}j∈J (see Definition 2.5 in [15]). A direct calculation can show that a dual K-Bessel sequence is necessarily a K∗-frame.

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

Example 6.

Suppose that H=C3, {gj}j=13={e1,e2,e3}, where (10)e1=100,e2=010,e3=001.Define K∈B(H) as follows: (11)K:H→H,Ke1=e1,Ke2=e1,Ke3=e2.Taking fj=Kgj for j=1,2,3, then (12)K∗f=∑j=13K∗f,ejej=∑j=13f,Kejej=∑j=13f,Kgjej=∑j=13f,fjej,for every f∈H. Therefore K∗f2=∑j=13f,fj2, showing that {fj}j=13 is a Parseval K-frame of H. Since (13)Kf=K∑j=13f,ejej=∑j=13f,ejKej=∑j=13f,gjfj,it follows that {gj}j=13 is a dual K-Bessel sequence of {fj}j=13. Let (14)h1=120,h2=0-10,h3=001,and then it is easy to check that {hj}j=13 is Bessel sequence of H. Now for any f∈H we have (15)∑j=13f,hjfj=f,e1+2e2e1+f,-e2e1+f,e3e2=f,e1+e2f1+f,e3f3=∑j=13f,gjfj=Kf.Hence {hj}j=13 is a dual K-Bessel sequence of {fj}j=13 and is different from {gj}j=13.

Guo in [15] proved that, among all dual K-Bessel sequences of a given K-frame, there is a unique dual K-Bessel sequence whose analysis operator obtains the minimal norm, which is called the canonical dual K-Bessel sequence. Motivated by the idea of [21], in the following we characterize the exact structure of the canonical dual K-Bessel sequence of a Parseval K-frame under the condition that K 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 K∈B(H) has closed range and {fj}j∈J is a Parseval K-frame of H, then {K†fj}j∈J is a dual K-Bessel sequence of {fj}j∈J.

Lemma 8.

Suppose that K∈B(H) has closed range and {fj}j∈J is a Parseval K-frame of H with analysis operator UF, then {gj}j∈J in H is a dual K-Bessel sequence of {fj}j∈J if and only if there exists φ∈B(H,l2(J)) such that UF∗φ=0 and 〈f,gj-K†fj〉=(φf)j for every f∈H and every j∈J.

By using Lemmas 7 and 8 we can obtain the following result which shows that the dual K-Bessel sequence {K†fj}j∈J of Parseval K-frame {fj}j∈J stated in Lemma 7 is exactly the canonical dual K-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 K∈B(H) has closed range and {fj}j∈J is a Parseval K-frame of H with analysis operator UF, then {K†fj}j∈J is the canonical dual K-Bessel sequence of {fj}j∈J.

Remark 10.

(1) The canonical dual K-Bessel sequence of the Parseval K-frame {fj}j∈J, which will be denoted by {f~j}j∈J later, is actually a Parseval frame on (KerK)⊥ since (16)∑j∈Jf,f~j2=K∗K†∗f2=K†K∗f2=K†Kf2=f2for every f∈(KerK)⊥, by Lemma 3.

(2) The canonical dual K-Bessel sequence of the Parseval K-frame {fj}j∈J is precisely a Parseval K†K-frame since (17)∑j∈Jf,f~j2=∑j∈Jf,K†fj2=K∗K†∗f2=K†K∗f2,∀f∈H.

(3) Although the canonical dual K-Bessel sequence of the Parseval K-frame {fj}j∈J is not a Parseval K-frame in general, it can naturally generate a new one in the form {Kf~j}j∈J. Indeed, by Lemma 3 we have (18)∑j∈Jf,Kf~j2=∑j∈Jf,KK†fj2=∑j∈JKK†∗f,fj2=K∗KK†∗f2=KK†K∗f2=K∗f2,∀f∈H.

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

Theorem 11.

Suppose that K∈B(H) has closed range and {fj}j∈J is a Parseval K-frame of H with a dual K-Bessel sequence {gj}j∈J. Then {gj}j∈J is the canonical dual K-Bessel sequence of {fj}j∈J if and only if UG∗UG=UG∗UH for any dual K-Bessel sequence {hj}j∈J of {fj}j∈J, where UG and UH denote the analysis operators of {gj}j∈J and {hj}j∈J, respectively.

Proof.

Let us first assume that {gj}j∈J={f~j}j∈J. If we denote by UF the analysis operator of {fj}j∈J, then a direct calculation can show that UG=UF(K†)∗. From this fact and taking into account the fact that (19)UF∗UGf-UHf=∑j∈Jf,f~jfj-∑j∈Jf,hjfj=0,∀f∈H,we obtain (20)UG-UHf,UGg=UG-UHf,UFK†∗g=K†UF∗UG-UHf,g=0for any f,g∈H. Thus UG∗(UG-UH)f=0; equivalently, UG∗UG=UG∗UH.

Conversely, let UG∗UG=UG∗UH for any dual K-Bessel sequence {hj}j∈J of {fj}j∈J. Then (21)UG2=UG∗UG=UG∗UH≤UGUH,and it follows that UG≤UH, implying that {gj}j∈J is the canonical dual K-Bessel sequence of {fj}j∈J. This completes the proof.

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

Theorem 12.

Suppose that K∈B(H) has closed range and {fj}j∈J is a Parseval K-frame of H with analysis operator UF. Then {fj}j∈J has a unique dual K-Bessel sequence if and only if Range(UF)=l2(J).

Proof.

Assume first that Range(UF)=l2(J). Then UF∗ is injective. Let {gj}j∈J and {hj}j∈J be two dual K-Bessel sequences of {fj}j∈J; then it is easy to check that {〈f,gj-hj〉}j∈J∈l2(J) for each f∈H and that (22)0=Kf-Kf=∑j∈Jf,gjfj-∑j∈Jf,hjfj=∑j∈Jf,gj-hjfj=UF∗f,gj-hjj∈J.Thus 〈f,gj-hj〉=0 for any j∈J and f∈H and, consequently, gj=hj for any j∈J.

For the opposite implication, assume contrarily that Range(UF)≠l2(J). Since {fj}j∈J is a Parseval K-frame, it is easily seen that KK∗=UF∗UF. Thus Range(UF∗)=Range(K) by Lemma 4, and UF has closed range as a consequence. Let S∈B(H,l2(J)) be an invertible operator and 0≠{aj}j∈J∈(Range(UF))⊥. Taking h=S-1({aj}j∈J) and gj=a¯jh for each j∈J, then, for every f∈H, (23)∑j∈Jf,gj2=∑j∈Jf,a¯jh2=∑j∈Jf,h2aj2=f,h2ajj∈J2≤ajj∈J2h2f2,meaning that {gj}j∈J is a Bessel sequence of H. Now let hj=f~j+gj for every j∈J; then it is easily seen that {hj}j∈J is a Bessel sequence of H. Since {aj}j∈J is orthogonal to Range(UF), (24)∑j∈Jf,gjfj,e=∑j∈Jf,a¯jhfj,e=∑j∈Jf,hajfj,e=f,hajj∈J,e,fjj∈J=f,hajj∈J,UFe=0for any e,f∈H. Therefore ∑j∈J〈f,gj〉fj=0 for any f∈H, which yields (25)∑j∈Jf,hjfj=∑j∈Jf,f~jfj+∑j∈Jf,gjfj=∑j∈Jf,f~jfj=Kf.Since {aj}j∈J≠0, there exists j0∈J such that aj0≠0 and, thus, gj0≠0, since a simple calculation gives aj0/|aj0|2S(gj0)={aj}j∈J. Hence {hj}j∈J is a dual K-Bessel sequence of {fj}j∈J and is different from {f~j}j∈J, a contradiction.

It is interesting that the l2(J)-linear independence of a Parseval K-frame {fj}j∈J can immediately lead to the l2(J)-linear independence of {f~j}j∈J and vice versa and that the uniqueness of the dual K-Bessel sequence of {fj}j∈J implies the uniqueness of the dual K∗-Bessel sequence of {f~j}j∈J.

Theorem 13.

Suppose that K∈B(H) has closed range and {fj}j∈J is a Parseval K-frame of H. Then the following results hold:

(1){fj}j∈J is l2(J)-linearly independent if and only if {f~j}j∈J is l2(J)-linearly independent.

(2) If {fj}j∈J admits a unique dual K-Bessel sequence, then {f~j}j∈J admits a unique dual K∗-Bessel sequence.

Proof.

(1) We first prove the necessity. Let UF be the analysis operator of {fj}j∈J; then (26)∑j∈Jf,fjfj=UF∗UFf=KK∗f=∑j∈JK∗f,f~jfj=∑j∈Jf,Kf~jfj,∀f∈H.Therefore,(27)0=∑j∈Jf,fjfj-∑j∈Jf,Kf~jfj=∑j∈Jf,fj-Kf~jfj.Since {fj}j∈J is l2(J)-linearly independent, it follows that 〈f,fj-Kf~j〉=0 for any f∈H and any j∈J, and fj=Kf~j for any j∈J as a consequence. Suppose now that ∑j∈Jcjf~j=0 for some {cj}j∈J∈l2(J), then (28)0=K∑j∈Jcjf~j=∑j∈JcjKf~j=∑j∈Jcjfj.Again by the l2(J)-linear independence of {fj}j∈J we obtain cj=0 for each j∈J.

For the sufficiency, let ∑j∈Jcjfj=0 for {cj}j∈J∈l2(J). Then (29)0=K†∑j∈Jcjfj=∑j∈JcjK†fj=∑j∈Jcjf~j.Thus cj=0 for every j∈J, and the conclusion follows.

(2) Since {fj}j∈J has a unique dual K-Bessel sequence, by Theorem 12 we know that its analysis operator UF is surjective and, thus, UF∗ is injective, which implies that {fj}j∈J is l2(J)-linearly independent. Hence, by (1), {f~j}j∈J is also l2(J)-linearly independent, from which we conclude that {f~j}j∈J has a unique dual K∗-Bessel sequence.

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

Theorem 14.

Suppose that K∈B(H) has closed range and {fj}j∈J is a Parseval K-frame of H. If the remainder of {fj}j∈J fails to be a new K-frame whenever any one of its elements is deleted, then {f~j}j∈J has a unique dual K∗-Bessel sequence.

Proof.

To prove that {f~j}j∈J admits a unique dual K∗-Bessel sequence, it is sufficient to show that {f~j}j∈J is l2(J)-linearly independent, and it is enough to show that {fj}j∈J is l2(J)-linearly independent by Theorem 13(1). Suppose on the contrary that there is {cj}j∈J∈l2(J) with cj0≠0 for some j0∈J such that ∑j∈Jcjfj=0. Then fj0=-∑j≠j0cj/cj0fj. Denote by UF the analysis operator of {fj}j∈J; then, by Lemma 5, Range(K)⊂Range(UF∗). Thus for each f∈Range(K) there exists {aj}j∈J∈l2(J) such that (30)f=UF∗ajj∈J=∑j∈Jajfj=∑j≠j0ajfj-aj0∑j≠j0cjcj0fj=∑j≠j0aj-aj0cj0cjfj.Clearly, {fj}j∈J\{j0} is a Bessel sequence of H and {aj-aj0/cj0cj}j∈J\{j0}∈l2(J\{j0}). Therefore, f∈Range(UFj0∗) and, consequently, Range(K)⊂Range(UFj0∗), where UFj0 is the analysis operator of {fj}j∈J\{j0}. Again by Lemma 5 it follows that {fj}j∈J\{j0} is a K-frame of H, which leads to a contradiction.

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

Theorem 15.

Suppose that K has closed range and {fj}j∈J is a Parseval K-frame of H. Then for any {cj}j∈J∈l2(J) satisfying Kf=∑j∈Jcjfj, we have (31)∑j∈Jcj2=∑j∈Jcj-f,f~j2+∑j∈Jf,f~j2.

Proof.

It is easy to check that (32)∑j∈Jcj-f,f~jf~j,f=∑j∈Jcj-f,f~jf~j,f=K†∑j∈Jcj-f,f~jfj,f=K†Kf-Kf,f=0for every f∈H. Therefore, (33)∑j∈Jcj2=∑j∈Jcjc¯j=∑j∈Jcj-f,f~j+f,f~jcj-f,f~j+f,f~j¯=∑j∈Jcj-f,f~jcj-f,f~j¯+cj-f,f~jf~j,f+f,f~jcj-f,f~j¯+f,f~jf~j,f=∑j∈Jcj-f,f~jcj-f,f~j¯+∑j∈Jf,f~jf~j,f=∑j∈Jcj-f,f~j2+∑j∈Jf,f~j2.

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).

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