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Some new results on the weaving of K-g-frames in Hilbert spaces

  • Zhong-Qi Xiang EMAIL logo
Published/Copyright: December 31, 2021
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Open Mathematics
From the journal Open Mathematics Volume 19 Issue 1

Abstract

In this paper, we provide some conditions for a K-woven pair of K-g-frames to be preserved under an operator and particularly, we report that applying two different operators to a K-woven pair of K-g-frames can leave them K-woven. Several new methods on the construction of K-woven pair of K-g-frames are also given. We end the paper with a new perturbation result on the weaving of K-g-frames, which shows that, under the perturbation condition involved in one known result on this topic, two K-g-frames can be K-woven in the whole space, not merely in the subspace Range(K).

Keywords: K-g-frame; K-woven; perturbation; pseudo-inverse
MSC 2010: 42C15; 47B40

1 Introduction

Inspired by the double inequalities induced from the frame coefficients of a multi-wavelet frame for L 2 ( R ) , Sun [1] introduced the notion of g-frames, which cover various generalizations of (classical) frames such as pseudo-frames, fusion frames, outer frames, and oblique frames. Even though many properties of frames have counterparts in g-frame setting, there are some essential differences between them. For example, an exact g-frame is not necessarily a g-Riesz basis, and not all g-Riesz frames contain a g-Riesz basis. This makes many scholars pay their attention to g-frames [2,3, 4,5].

Later, K-frames, also known as frames for operators, were put forward by Găvruţa in [6] to examine atomic decompositions for a linear bounded operator K , which, in many aspects, behave quite differently to frames, see [7,8,9, 10,11]. After those works, Xiao et al. [12] showed us a more general type of frames named K-g-frames by combing g-frames with K-frames. Asgari and Rahimi in [13] called these frames “Generalized frames for operators.” The readers can check [14,15,16] for more details on K-g-frames.

Motivated by a question arising in distributed signal processing, Bemrose et al. [17,18] proposed the concept of weaving frames, which were further investigated in [19,20]. Now, weaving has become to be a research hotspot because of some potential applications. We refer to the papers [21,22,23] for more information on the weaving of fusion frames or g-frames, and the paper [24] for information on the weaving of K-frames.

Recently, the concept of weaving for more general frames, namely weaving K-g-frames, was introduced in [25], which provides a wider perspective on frame theory, and a deep study of them would help to explore the application approach of weaving theory of frames in pre-processing of signals and wireless sensor networks and some other fields. So it is natural for us to come up with the idea to examine this new framework, in order to further develop and enrich frame theory and its application scenarios.

In [25], the authors studied the erasures and perturbations of weaving for K-g-frames. We found, however, that the main results they obtained are all based on the subspace Range ( K ) . The aim of this article is to present some new results on the weaving for K-g-frames, which are established in the whole space. More precisely, in Section 2 we examine conditions under which a K-woven pair of K-g-frames can be preserved when applying linear bounded operators to them. Section 3 is devoted to constructing K-woven pair of K-g-frames by using several new methods. In Section 4, we give a new perturbation result on the weaving of K-g-frames, which shows that the Paley-Wiener-type perturbation condition involved in Theorem 3 of [25] is valid for the whole space.

Throughout this paper, I is an index set (finite or countable), H , K , and K i ’s ( i ∈ I ) are complex separable Hilbert spaces. We denote by B ( H , K ) the collection of all linear bounded operators from H to K , and B ( H , H ) is abbreviated to B ( H ) .

Definition 1.1

(see [12]) Suppose K ∈ B ( H ) and Λ i ∈ B ( H , K i ) for each i ∈ I . We say that { Λ i } i ∈ I is a K-g-frame for H with respect to { K i } i ∈ I with lower and upper frame bounds C and D , respectively, if there are constants C , D > 0 such that

(1.1) C ‖ K ∗ f ‖ 2 ≤ ∑ i ∈ I ‖ Λ i f ‖ 2 ≤ D ‖ f ‖ 2 , ∀ f ∈ H .

Recall that a K-g-frame { Λ i } i ∈ I is said to be C -tight, if

C ‖ K ∗ f ‖ 2 = ∑ i ∈ I ‖ Λ i f ‖ 2 , ∀ f ∈ H .

If only the inequality to the right-hand side in (1.1) holds, then { Λ i } i ∈ I is called a g-Bessel sequence with g-Bessel bound D , related to which there are two well-defined operators:

T : ℓ 2 ( { K i } i ∈ I ) → H , T { g i } i ∈ I = ∑ i ∈ I Λ i ∗ g i ( synthesis operator ) S : H → H , S f = ∑ i ∈ I Λ i ∗ Λ i f ( g-frame operator )

Definition 1.2

(see [25]) Let { Λ i } i ∈ I and { Γ i } i ∈ I be K-g-frames for H with respect to { K i } i ∈ I . If there are universal constants C and D such that for any partition { σ 1 , σ 2 } of I , { Λ i } i ∈ σ 1 ∪ { Γ i } i ∈ σ 2 is a K-g-frame for H with frame bounds C and D , then we call that { Λ i } i ∈ I and { Γ i } i ∈ I are K-woven in H , and each { Λ i } i ∈ σ 1 ∪ { Γ i } i ∈ σ 2 is called a weaving.

The K-g-frames { Λ i } i ∈ I and { Γ i } i ∈ I are said to be C -tight K-woven, if for any partition { σ 1 , σ 2 } of I , { Λ i } i ∈ σ 1 ∪ { Γ i } i ∈ σ 2 is a C -tight K-g-frame.

Note that if K = Id H , the identical operator on H , then the weaving of K-g-frames turns to be the weaving of g-frames.

Lemma 1.3

(see [26]) Suppose that Φ ∈ B ( H , K ) has closed range, then there exists a unique operator Φ † ∈ B ( K , H ) , called the pseudo-inverse of Φ , satisfying

Φ Φ † = P Range ( Φ ) , Φ † Φ = P Range ( Φ † ) , ( Φ Φ † ) ∗ = Φ Φ † , ( Φ † Φ ) ∗ = Φ † Φ , Ker Φ † = ( Range ( Φ ) ) ⊥ , Range ( Φ † ) = ( Ker Φ ) ⊥ ,

where P Range ( Φ ) denotes the orthogonal projection onto Range ( Φ ) .

Lemma 1.4

(see [27]) Let H 1 , H 2 be two Hilbert spaces. Also let P ∈ B ( H 1 , H ) and Q ∈ B ( H 2 , H ) . The following statements are equivalent.

  1. Range ( P ) ⊂ Range ( Q ) .

  2. There exists λ > 0 such that P P ∗ ≤ λ Q Q ∗ .

  3. There exists Θ ∈ B ( H 1 , H 2 ) such that P = Q Θ .

In the sequel, the notations D Λ , T Λ , and S Λ are used to denote, respectively, the g-Bessel bound, the synthesis operator, and the g-frame operator of the g-Bessel sequence { Λ i } i ∈ I (the same goes for { Γ i } i ∈ I and { Θ i } i ∈ I ) and if, furthermore, { Λ i } i ∈ I is a K-g-frame, then we use C Λ to denote the lower frame bound (the same goes for { Γ i } i ∈ I ). As usual, the notation Φ † is reserved for the pseudo-inverse of Φ , if it exists.

2 Operator perturbation of the weaving of K-g-frames

The main purpose of this section is to investigate the stability of K-woven pair of K-g-frames under operator perturbation.

Theorem 2.1

Suppose that K-g-frames { Λ i } i ∈ I and { Γ i } i ∈ I are K-woven in H with the universal frame bounds C and D . If U ∈ B ( H ) has closed range with U K = K U and Range ( K ∗ ) ⊂ Range ( U ) , then { Λ i U ∗ } i ∈ I and { Γ i U ∗ } i ∈ I are K-woven in H with the universal frame bounds C ‖ U † ‖ − 2 and D ‖ U ‖ 2 .

Proof

For each f ∈ H we have

∑ i ∈ I ‖ Λ i U ∗ f ‖ 2 ≤ D Λ ‖ U ∗ f ‖ 2 ≤ D Λ ‖ U ‖ 2 ‖ f ‖ 2 .

Since U K = K U , it follows that K ∗ U ∗ = U ∗ K ∗ . Noting that U has closed range and that Range ( K ∗ ) ⊂ Range ( U ) , by Lemma 1.3 we get

‖ K ∗ f ‖ 2 = ‖ U U † K ∗ f ‖ 2 = ‖ ( U † ) ∗ K ∗ U ∗ f ‖ 2 ≤ ‖ U † ‖ 2 ‖ K ∗ U ∗ f ‖ 2 , ∀ f ∈ H ,

which tells us that

∑ i ∈ I ‖ Λ i U ∗ f ‖ 2 ≥ C Λ ‖ K ∗ U ∗ f ‖ 2 ≥ C Λ ‖ U † ‖ − 2 ‖ K ∗ f ‖ 2 .

Therefore, { Λ i U ∗ } i ∈ I is a K-g-frame for H with frame bounds C Λ ‖ U † ‖ − 2 and D Λ ‖ U ‖ 2 . Similarly, we can show that { Γ i U ∗ } i ∈ I is a K-g-frame for H with frame bounds C Γ ‖ U † ‖ − 2 and D Γ ‖ U ‖ 2 .

Since { Λ i } i ∈ σ 1 ∪ { Γ i } i ∈ σ 2 is a K-g-frame for H with frame bounds C and D for any partition { σ 1 , σ 2 } of I , it follows that { Λ i U ∗ } i ∈ σ 1 ∪ { Γ i U ∗ } i ∈ σ 2 is a K-g-frame for H with frame bounds C ‖ U † ‖ − 2 and D ‖ U ‖ 2 . That is, { Λ i U ∗ } i ∈ I and { Γ i U ∗ } i ∈ I are K-woven in H with the universal frame bounds C ‖ U † ‖ − 2 and D ‖ U ‖ 2 .□

Corollary 2.2

Suppose that K ∈ B ( H ) has a dense range, that { Λ i } i ∈ I and { Γ i } i ∈ I are K-g-frames for H with respect to { K i } i ∈ I , and that U ∈ B ( H ) has closed range with U K = K U . If { Λ i U ∗ } i ∈ I and { Γ i U ∗ } i ∈ I , and { Λ i U } i ∈ I and { Γ i U } i ∈ I are both K-woven in H , then { Λ i } i ∈ I and { Γ i } i ∈ I are K-woven in H .

Proof

Let C and D be the universal frame bounds of { Λ i U ∗ } i ∈ I and { Γ i U ∗ } i ∈ I . Then for any partition { σ 1 , σ 2 } of I ,

(2.1) C ‖ K ∗ f ‖ 2 ≤ ∑ i ∈ σ 1 ‖ Λ i U ∗ f ‖ 2 + ∑ i ∈ σ 2 ‖ Γ i U ∗ f ‖ 2 , ∀ f ∈ H .

Since K has a dense range, Range ¯ ( K ) = H , and Ker ( K ∗ ) = ( Range ¯ ( K ) ) ⊥ = { 0 } as a consequence. This together with (2.1) gives Range ( U ) = ( Ker ( U ∗ ) ) ⊥ ⊃ ( Ker ( K ∗ ) ) ⊥ = H , meaning that U is surjective. A similar discussion can show that U ∗ is also surjective. Hence, U is invertible on H . Since U K = K U , U − 1 K = K U − 1 . Now the conclusion follows from Theorem 2.1, since { Λ i } i ∈ I = { Λ i U ∗ ( U − 1 ) ∗ } i ∈ I and { Γ i } i ∈ I = { Γ i U ∗ ( U − 1 ) ∗ } i ∈ I , and Range ( K ∗ ) ⊂ H = Range ( U − 1 ) .□

Corollary 2.3

Let K-g-frames { Λ i } i ∈ I and { Γ i } i ∈ I be K-woven in H and U ∈ B ( H ) with U K = K U . If U is surjective, then { Λ i U ∗ } i ∈ I and { Γ i U ∗ } i ∈ I are K-woven in H .

Proof

It is an immediate consequence of Theorem 2.1.□

In fact, the inverse of Corollary 2.3 remains true if the K-woven is tight and K ∗ is surjective, as shown in the following result.

Theorem 2.4

Suppose that K-g-frames { Λ i } i ∈ I and { Γ i } i ∈ I are C -tight K-woven in H , that U ∈ B ( H ) with U K = K U , and that Range ( K ∗ ) = H . If { Λ i U ∗ } i ∈ I and { Γ i U ∗ } i ∈ I are K-woven in H , then U is surjective.

Proof

Let C ′ and D ′ be the universal frame bounds of { Λ i U ∗ } i ∈ I and { Γ i U ∗ } i ∈ I . Then for any partition { σ 1 , σ 2 } of I and any f ∈ H ,

(2.2) C ′ ‖ K ∗ f ‖ 2 ≤ ∑ i ∈ σ 1 ‖ Λ i U ∗ f ‖ 2 + ∑ i ∈ σ 2 ‖ Γ i U ∗ f ‖ 2 ≤ D ′ ‖ f ‖ 2 .

We also have

C ‖ K ∗ g ‖ 2 = ∑ i ∈ σ 1 ‖ Λ i g ‖ 2 + ∑ i ∈ σ 2 ‖ Γ i g ‖ 2

for any g ∈ H , and the condition U K = K U yields U ∗ K ∗ = K ∗ U ∗ . Thus,

C ‖ U ∗ K ∗ f ‖ 2 = C ‖ K ∗ U ∗ f ‖ 2 = ∑ i ∈ σ 1 ‖ Λ i U ∗ f ‖ 2 + ∑ i ∈ σ 2 ‖ Γ i U ∗ f ‖ 2 .

This fact along with (2.2) implies that

‖ U ∗ K ∗ f ‖ 2 = C − 1 ∑ i ∈ σ 1 ‖ Λ i U ∗ f ‖ 2 + ∑ i ∈ σ 2 ‖ Γ i U ∗ f ‖ 2 ≥ C − 1 C ′ ‖ K ∗ f ‖ 2 ,

from which we conclude that U ∗ is bounded below as K ∗ is surjective and, consequently, U is surjective.□

With a similar discussion to Theorem 2.4 we can obtain the following result.

Proposition 2.5

Suppose that K-g-frames { Λ i } i ∈ I and { Γ i } i ∈ I are 1-tight K-woven in H , that U ∈ B ( H ) with U K = K U , and that Range ( K ∗ ) = H . If { Λ i U ∗ } i ∈ I and { Γ i U ∗ } i ∈ I are 1-tight K-woven in H , then U is a co-isometry.

Furthermore, we report in the following result that two different operators are applied to a K-woven pair of K-g-frames can also leave them K-woven.

Theorem 2.6

Suppose that K-g-frames { Λ i } i ∈ I and { Γ i } i ∈ I are K-woven in H with the universal frame bounds C and D , that U i ∈ B ( H ) are surjective with U i K = K U i ( i = 1 , 2 ). If Ker ( K ∗ ) ⊂ Ker ( U i ∗ ) for i = 1 , 2 , and ‖ U 1 † ‖ ‖ U 2 − U 1 ‖ ‖ K ∗ ˜ − 1 ‖ < C / D Γ , then { Λ i U 1 ∗ } i ∈ I and { Γ i U 2 ∗ } i ∈ I are K-woven in H with the universal frame bounds ( C ‖ U 1 † ‖ − 1 − D Γ ‖ U 2 − U 1 ‖ ‖ K ∗ ˜ − 1 ‖ ) 2 and D Λ ‖ U 1 ‖ 2 + D Γ ‖ U 2 ‖ 2 , where K ∗ ˜ : Ker ⊥ ( K ∗ ) → Range ( K ∗ ) is the restriction of K ∗ on Ker ⊥ ( K ∗ ) .

Proof

The proof of Theorem 2.1 tells us that both { Λ i U 1 ∗ } i ∈ I and { Γ i U 2 ∗ } i ∈ I are K-g-frames for H . Now for any partition { σ 1 , σ 2 } of I and any g ∈ Ker ⊥ ( K ∗ ) we have

∑ i ∈ σ 1 ‖ Λ i U 1 ∗ g ‖ 2 + ∑ i ∈ σ 2 ‖ Γ i U 2 ∗ g ‖ 2 1 / 2 = ∑ i ∈ σ 1 ‖ Λ i U 1 ∗ g ‖ 2 + ∑ i ∈ σ 2 ‖ Γ i U 1 ∗ g + Γ i ( U 2 ∗ − U 1 ∗ ) g ‖ 2 1 / 2 ≥ ∑ i ∈ σ 1 ‖ Λ i U 1 ∗ g ‖ 2 + ∑ i ∈ σ 2 ‖ Γ i U 1 ∗ g ‖ 2 1 / 2 − ∑ i ∈ σ 2 ‖ Γ i ( U 2 ∗ − U 1 ∗ ) g ‖ 2 1 / 2

≥ C ‖ K ∗ U 1 ∗ g ‖ − ∑ i ∈ I ‖ Γ i ( U 2 ∗ − U 1 ∗ ) g ‖ 2 1 / 2 ≥ C ‖ K ∗ U 1 ∗ g ‖ − D Γ ‖ U 2 ∗ − U 1 ∗ ‖ ‖ g ‖ = C ‖ U 1 ∗ K ∗ g ‖ − D Γ ‖ U 2 − U 1 ‖ ‖ K ∗ ˜ − 1 K ∗ ˜ g ‖ ≥ C ‖ U 1 † ‖ − 1 ‖ K ∗ g ‖ − D Γ ‖ U 2 − U 1 ‖ ‖ K ∗ ˜ − 1 ‖ ‖ K ∗ ˜ g ‖ = ( C ‖ U 1 † ‖ − 1 − D Γ ‖ U 2 − U 1 ‖ ‖ K ∗ ˜ − 1 ‖ ) ‖ K ∗ g ‖ .

Since each f ∈ H can be represented as f = f 1 + f 2 with f 1 ∈ Ker ( K ∗ ) and f 2 ∈ Ker ⊥ ( K ∗ ) , and Ker ( K ∗ ) ⊂ Ker ( U i ∗ ) ( i = 1 , 2 ), it follows that

∑ i ∈ σ 1 ‖ Λ i U 1 ∗ f ‖ 2 + ∑ i ∈ σ 2 ‖ Γ i U 2 ∗ f ‖ 2 = ∑ i ∈ σ 1 ‖ Λ i U 1 ∗ ( f 1 + f 2 ) ‖ 2 + ∑ i ∈ σ 2 ‖ Γ i U 2 ∗ ( f 1 + f 2 ) ‖ 2 = ∑ i ∈ σ 1 ‖ Λ i U 1 ∗ f 2 ‖ 2 + ∑ i ∈ σ 2 ‖ Γ i U 2 ∗ f 2 ‖ 2 ≥ ( C ‖ U 1 † ‖ − 1 − D Γ ‖ U 2 − U 1 ‖ ‖ K ∗ ˜ − 1 ‖ ) 2 ‖ K ∗ f 2 ‖ 2 = ( C ‖ U 1 † ‖ − 1 − D Γ ‖ U 2 − U 1 ‖ ‖ K ∗ ˜ − 1 ‖ ) 2 ‖ K ∗ ( f 1 + f 2 ) ‖ 2 = ( C ‖ U 1 † ‖ − 1 − D Γ ‖ U 2 − U 1 ‖ ‖ K ∗ ˜ − 1 ‖ ) 2 ‖ K ∗ f ‖ 2 .

The required universal upper frame bound follows from the following calculation:

∑ i ∈ σ 1 ‖ Λ i U 1 ∗ f ‖ 2 + ∑ i ∈ σ 2 ‖ Γ i U 2 ∗ f ‖ 2 ≤ ∑ i ∈ I ‖ Λ i U 1 ∗ f ‖ 2 + ∑ i ∈ I ‖ Γ i U 2 ∗ f ‖ 2 ≤ ( D Λ ‖ U 1 ‖ 2 + D Γ ‖ U 2 ‖ 2 ) ‖ f ‖ 2 , ∀ f ∈ H . □

At the end of this section we discuss conditions under which the sum of a K-woven pair of K-g-frames and their perturbation under an operator can be still K-woven.

Theorem 2.7

Suppose that K-g-frames { Λ i } i ∈ I and { Γ i } i ∈ I are K-woven in H with the universal frame bounds C and D and that T ∈ B ( H ) is a positive operator. For any σ ⊂ I , let S σ be the g-frame operator of { Λ i } i ∈ σ ∪ { Γ i } i ∈ σ c . If T S σ = S σ T , then { Λ i + Λ i T ∗ } i ∈ I and { Γ i + Γ i T ∗ } i ∈ I are K-woven in H with the universal frame bounds C and D ‖ Id H + T ‖ 2 .

Proof

For any partition { σ 1 , σ 2 } of I and any f ∈ H we have

∑ i ∈ σ 1 ‖ ( Λ i + Λ i T ∗ ) f ‖ 2 + ∑ i ∈ σ 2 ‖ ( Γ i + Γ i T ∗ ) f ‖ 2 = ∑ i ∈ σ 1 ‖ Λ i ( Id H + T ∗ ) f ‖ 2 + ∑ i ∈ σ 2 ‖ Γ i ( Id H + T ∗ ) f ‖ 2 ≤ D ‖ ( Id H + T ) ∗ f ‖ 2 ≤ D ‖ Id H + T ‖ 2 ‖ f ‖ 2 .

Since T is a positive operator with T S σ 1 = S σ 1 T , we can get T S σ 1 ≥ 0 and S σ 1 T ∗ ≥ 0 . If, now, we denote by S the g-frame operator of { Λ i + Λ i T ∗ } i ∈ σ 1 ∪ { Γ i + Γ i T ∗ } i ∈ σ 2 , then

S f = ∑ i ∈ σ 1 ( Λ i + Λ i T ∗ ) ∗ ( Λ i + Λ i T ∗ ) f + ∑ i ∈ σ 2 ( Γ i + Γ i T ∗ ) ∗ ( Γ i + Γ i T ∗ ) f = ∑ i ∈ σ 1 Λ i ∗ Λ i f + ∑ i ∈ σ 2 Γ i ∗ Γ i f + T ∑ i ∈ σ 1 Λ i ∗ Λ i f + ∑ i ∈ σ 2 Γ i ∗ Γ i f + ∑ i ∈ σ 1 Λ i ∗ Λ i T ∗ f + ∑ i ∈ σ 2 Γ i ∗ Γ i T ∗ f + T ∑ i ∈ σ 1 Λ i ∗ Λ i T ∗ f + ∑ i ∈ σ 2 Γ i ∗ Γ i T ∗ f = S σ 1 f + T S σ 1 f + S σ 1 T ∗ f + T S σ 1 T ∗ f ≥ S σ 1 f

for any f ∈ H . Therefore,

∑ i ∈ σ 1 ‖ ( Λ i + Λ i T ∗ ) f ‖ 2 + ∑ i ∈ σ 2 ‖ ( Γ i + Γ i T ∗ ) f ‖ 2 = ⟨ S f , f ⟩ ≥ ⟨ S σ 1 f , f ⟩ ≥ C ‖ K ∗ f ‖ 2 ,

and we have the result.□

Theorem 2.8

Suppose that K-g-frames { Λ i } i ∈ I and { Γ i } i ∈ I are K-woven in H with the universal frame bounds C and D and that T ∈ B ( H ) is a positive operator. If T K K ∗ is positive, then { Λ i + Λ i T ∗ } i ∈ I and { Γ i + Γ i T ∗ } i ∈ I are K-woven in H with the universal frame bounds C and D ‖ Id H + T ‖ 2 .

Proof

For each f ∈ H we have

(2.3) ∑ i ∈ I ‖ ( Λ i + Λ i T ∗ ) f ‖ 2 = ∑ i ∈ I ‖ Λ i ( Id H + T ∗ ) f ‖ 2 ≤ D Λ ‖ ( Id H + T ) ∗ f ‖ 2 ≤ D Λ ‖ Id H + T ‖ 2 ‖ f ‖ 2 .

Now, noting that T K K ∗ is a positive operator, we obtain

∑ i ∈ I ‖ ( Λ i + Λ i T ∗ ) f ‖ 2 ≥ C Λ ‖ K ∗ ( Id H + T ∗ ) f ‖ 2 = C Λ ( ‖ K ∗ f ‖ 2 + 2 Re ⟨ K ∗ f , K ∗ T ∗ f ⟩ + ‖ K ∗ T ∗ f ‖ 2 ) = C Λ ( ‖ K ∗ f ‖ 2 + 2 Re ⟨ T K K ∗ f , f ⟩ + ‖ K ∗ T ∗ f ‖ 2 ) ≥ C Λ ( ‖ K ∗ f ‖ 2 + ‖ K ∗ T ∗ f ‖ 2 ) ≥ C Λ ‖ K ∗ f ‖ 2 .

This together with (2.3) means that { Λ i + Λ i T ∗ } i ∈ I is a K-g-frame for H with frame bounds C Λ and D Λ ‖ Id H + T ‖ 2 . Similarly, we know that { Γ i + Γ i T ∗ } i ∈ I is also a K-g-frame for H with frame bounds C Γ and D Γ ‖ Id H + T ‖ 2 .

Since { Λ i } i ∈ σ 1 ∪ { Γ i } i ∈ σ 2 is a K-g-frame for H with frame bounds C and D for any partition { σ 1 , σ 2 } of I , following above discussion we conclude that { Λ i + Λ i T ∗ } i ∈ σ 1 ∪ { Γ i + Γ i T ∗ } i ∈ σ 2 is a K-g-frame for H with frame bounds C and D ‖ Id H + T ‖ 2 . That is, { Λ i + Λ i T ∗ } i ∈ I and { Γ i + Γ i T ∗ } i ∈ I are K-woven in H with the universal frame bounds C and D ‖ Id H + T ‖ 2 .□

3 Constructions of the weaving of K-g-frames

In this section, we provide several new methods to construct K-woven pair of K-g-frames. We begin with a necessary and sufficient condition for two K-g-frames to be K-woven.

Theorem 3.1

Let { Λ i } i ∈ I and { Γ i } i ∈ I be two K-g-frames for H with respect to { K i } i ∈ I . Then { Λ i } i ∈ I and { Γ i } i ∈ I are K-woven in H if and only if there exists a g-Bessel sequence { Θ i } i ∈ I for H with respect to { K i } i ∈ I such that for any partition { σ 1 , σ 2 } of I and any f ∈ H ,

K f = ∑ i ∈ σ 1 Λ i ∗ Θ i f + ∑ i ∈ σ 2 Γ i ∗ Θ i f .

Proof

“ ⇐ .” It is sufficient to show the universal lower frame bound condition. For any f , g ∈ H ,

⟨ K g , f ⟩ = ∑ i ∈ σ 1 ⟨ Λ i ∗ Θ i g , f ⟩ + ∑ i ∈ σ 2 ⟨ Γ i ∗ Θ i g , f ⟩ = g , ∑ i ∈ σ 1 Θ i ∗ Λ i f + g , ∑ i ∈ σ 2 Θ i ∗ Γ i f = g , ∑ i ∈ σ 1 Θ i ∗ Λ i f + ∑ i ∈ σ 2 Θ i ∗ Γ i f ,

implying that K ∗ f = ∑ i ∈ σ 1 Θ i ∗ Λ i f + ∑ i ∈ σ 2 Θ i ∗ Γ i f , and thus

‖ K ∗ f ‖ 2 = sup ‖ h ‖ = 1 ∣ ⟨ K ∗ f , h ⟩ ∣ 2 = sup ‖ h ‖ = 1 ∑ i ∈ σ 1 ⟨ Λ i f , Θ i h ⟩ + ∑ i ∈ σ 2 ⟨ Γ i f , Θ i h ⟩ 2 ≤ 2 sup ‖ h ‖ = 1 ∑ i ∈ σ 1 ⟨ Λ i f , Θ i h ⟩ 2 + ∑ i ∈ σ 2 ⟨ Γ i f , Θ i h ⟩ 2 ≤ 2 D Θ ∑ i ∈ σ 1 ‖ Λ i f ‖ 2 + ∑ i ∈ σ 2 ‖ Γ i f ‖ 2 .

Therefore,

1 2 D Θ ‖ K ∗ f ‖ 2 ≤ ∑ i ∈ σ 1 ‖ Λ i f ‖ 2 + ∑ i ∈ σ 2 ‖ Γ i f ‖ 2 ,

as desired.

“ ⇒ .” Let C be the universal lower frame bound of { Λ i } i ∈ I and { Γ i } i ∈ I . For any partition { σ 1 , σ 2 } of I , let T be the synthesis operator of the g-Bessel sequence { Λ i } i ∈ σ 1 ∪ { Γ i } i ∈ σ 2 , and P n be the projection on ℓ 2 ( { K i } j ∈ I ) that maps each element to its n th component, i.e., p n { g i } i ∈ I = { u i } i ∈ I , where

u i = g n if i = n , 0 if i ≠ n ,

for each { g i } i ∈ I ∈ ℓ 2 ( { K i } i ∈ I ) . Since

C ⟨ K K ∗ f , f ⟩ ≤ ∑ i ∈ σ 1 ‖ Λ i f ‖ 2 + ∑ i ∈ σ 2 ‖ Γ i f ‖ 2 = ⟨ T T ∗ f , f ⟩ , ∀ f ∈ H ,

we know that K K ∗ ≤ 1 C T T ∗ . By Lemma 1.4, there exists U ∈ B ( H , ℓ 2 ( { K i } i ∈ I ) ) such that K = T U . Taking Θ i = P i U for i ∈ I , then

∑ i ∈ I ‖ Θ i f ‖ 2 = ∑ i ∈ I ‖ P i U f ‖ 2 = ∑ i ∈ I ‖ ( U f ) i ‖ 2 = ‖ U f ‖ 2 ≤ ‖ U ‖ 2 ‖ f ‖ 2

for any f ∈ H , showing that { Θ i } i ∈ I is a g-Bessel sequence for H with respect to { K i } i ∈ I . Now

K f = T U f = ∑ i ∈ σ 1 Λ i ∗ ( U f ) i + ∑ i ∈ σ 2 Γ i ∗ ( U f ) i = ∑ i ∈ σ 1 Λ i ∗ ( P i U f ) + ∑ i ∈ σ 2 Γ i ∗ ( P i U f ) = ∑ i ∈ σ 1 Λ i ∗ Θ i f + ∑ i ∈ σ 2 Γ i ∗ Θ i f . □

Corollary 3.2

Suppose that K ∈ B ( H ) has closed range and that Λ i , Γ i ∈ B ( H , K i ) for each i ∈ I . If { Λ i } i ∈ I and { Γ i } i ∈ I are a K-woven pair of K-g-frames in H , then they are a woven pair of g-frames in Range ( K ) .

Proof

Let { Λ i } i ∈ I and { Γ i } i ∈ I be a K-woven pair of K-g-frames in H . Then, by Theorem 3.1 there is a g-Bessel sequence { Θ i } i ∈ I for H with respect to { K i } i ∈ I such that for any partition { σ 1 , σ 2 } of I and any f ∈ H we have

K f = ∑ i ∈ σ 1 Λ i ∗ Θ i f + ∑ i ∈ σ 2 Γ i ∗ Θ i f .

For each h ∈ Range ( K ) , we see from Lemma 1.3 that

h = K K † h = ∑ i ∈ σ 1 Λ i ∗ ( Θ i K † P Range ( K ) ) h + ∑ i ∈ σ 2 Γ i ∗ ( Θ i K † P Range ( K ) ) h .

Taking Φ i = Θ i K † P Range ( K ) for any i ∈ I , then it is easy to check that { Φ i } i ∈ I is a g-Bessel sequence for H with bound D Θ ‖ K † ‖ 2 . Now for any g , h ∈ Range ( K ) ,

⟨ g , h ⟩ = ∑ i ∈ σ 1 Λ i ∗ Φ i g + ∑ i ∈ σ 2 Γ i ∗ Φ i g , h = ∑ i ∈ σ 1 ⟨ Λ i ∗ Φ i g , h ⟩ + ∑ i ∈ σ 2 ⟨ Γ i ∗ Φ i g , h ⟩ = ∑ i ∈ σ 1 ⟨ g , Φ i ∗ Λ i h ⟩ + ∑ i ∈ σ 2 ⟨ g , Φ i ∗ Γ i h ⟩ = g , ∑ i ∈ σ 1 Φ i ∗ Λ i h + ∑ i ∈ σ 2 Φ i ∗ Γ i h .

It follows that h = ∑ i ∈ σ 1 Φ i ∗ Λ i h + ∑ i ∈ σ 2 Φ i ∗ Γ i h . Hence,

‖ h ‖ 2 = sup ‖ g ‖ = 1 ∣ ⟨ h , g ⟩ ∣ 2 = sup ‖ g ‖ = 1 ∑ i ∈ σ 1 ⟨ Λ i h , Φ i g ⟩ + ∑ i ∈ σ 2 ⟨ Γ i h , Φ i g ⟩ 2 ≤ 2 sup ‖ g ‖ = 1 ∑ i ∈ σ 1 ⟨ Λ i h , Φ i g ⟩ 2 + ∑ i ∈ σ 2 ⟨ Γ i h , Φ i g ⟩ 2 ≤ 2 D Θ ‖ K † ‖ 2 ∑ i ∈ σ 1 ‖ Λ i h ‖ 2 + ∑ i ∈ σ 2 ‖ Γ i h ‖ 2 .

The universal upper frame bound condition is trivial, and the proof is complete.□

Corollary 3.3

Let { Λ i } i ∈ I and { Γ i } i ∈ I be two K-g-frames for H with respect to { K i } i ∈ I . Then { Λ i } i ∈ I and { Γ i } i ∈ I are K-woven in H if and only if there exists M > 0 such that for each f ∈ H , there is { c i f } i ∈ I ∈ ℓ 2 ( { K i } i ∈ I ) satisfying

  1. K f = ∑ i ∈ σ 1 Λ i ∗ c i f + ∑ i ∈ σ 2 Γ i ∗ c i f for any partition { σ 1 , σ 2 } of I , and

  2. ∑ i ∈ I ‖ c i f ‖ 2 ≤ M ‖ f ‖ 2 .

Proof

“ ⇒ .” By Theorem 3.1 there is a g-Bessel sequence { Θ i } i ∈ I for H with respect to { K i } i ∈ I such that K f = ∑ i ∈ σ 1 Λ i ∗ Θ i f + ∑ i ∈ σ 2 Γ i ∗ Θ i f for any partition { σ 1 , σ 2 } of I and any f ∈ H . Taking M = D Θ and c i f = Θ i f for any i ∈ I , then

K f = ∑ i ∈ σ 1 Λ i ∗ c i f + ∑ i ∈ σ 2 Γ i ∗ c i f and ∑ i ∈ I ‖ c i f ‖ 2 = ∑ i ∈ I ‖ Θ i f ‖ 2 ≤ D Θ ‖ f ‖ 2 = M ‖ f ‖ 2 .

“ ⇐ .” The proof is similar to the “ ⇐ ” part of Theorem 3.1, we omit the details.□

In next two theorems, we show that the weaving of K-g-frames can be checked in subspaces.

Theorem 3.4

Suppose that K-g-frames { Λ i } i ∈ I and { Γ i } i ∈ I are K-woven in M , a closed subspace of H , with the universal frame bounds C and D .

  1. If Range ( K ) ⊂ M , then { Λ i P M } i ∈ I and { Γ i P M } i ∈ I are K-woven in H with the same universal frame bounds.

  2. If M is dense in H , then { Λ i } i ∈ I and { Γ i } i ∈ I are K-woven in H with the same universal frame bounds.

Proof

  1. For any f ∈ H , P M f ∈ M and thus,

    ∑ i ∈ I ‖ Λ i P M f ‖ 2 ≤ D Λ ‖ P M f ‖ 2 ≤ D Λ ‖ f ‖ 2 .

Noting now that each f ∈ H has a decomposition f = f 1 + f 2 with f 1 ∈ M and f 2 ∈ M ⊥ , and that ⟨ K ∗ f 2 , g ⟩ = ⟨ f 2 , K g ⟩ = 0 for any g ∈ H leading to K ∗ f 2 = 0 , we obtain

∑ i ∈ I ‖ Λ i P M f ‖ 2 ≥ C Λ ‖ K ∗ P M f ‖ 2 = C Λ ‖ K ∗ P M ( f 1 + f 2 ) ‖ 2 = C Λ ‖ K ∗ f 1 ‖ 2 = C Λ ‖ K ∗ ( f 1 + f 2 ) ‖ 2 = C Λ ‖ K ∗ f ‖ 2 .

Altogether we know that { Λ i P M } i ∈ I is a K-g-frame for H with respect to { K i } i ∈ I with frame bounds C Λ and D Λ and, similarly, we can show that { Γ i P M } i ∈ I is a K-g-frame for H with respect to { K i } i ∈ I with frame bounds C Γ and D Γ .

Since { Λ i } i ∈ σ 1 ∪ { Γ i } i ∈ σ 2 is a K-g-frame for M with respect to { K i } i ∈ I with frame bounds C and D for any partition { σ 1 , σ 2 } of I , from above discussion we know that { Λ i P M } i ∈ σ 1 ∪ { Γ i P M } i ∈ σ 2 is a K-g-frame for H with respect to { K i } i ∈ I with frame bounds C and D . That is, { Λ i P M } i ∈ I and { Γ i P M } i ∈ I are K-woven in H with the universal frame bounds C and D .

  1. For any g ∈ M we have

    (3.1) C Λ ‖ K ∗ g ‖ 2 ≤ ∑ i ∈ I ‖ Λ i g ‖ 2 ≤ D Λ ‖ g ‖ 2 .

We prove first that { Λ i } i ∈ I is a g-Bessel sequence for H with respect to { K i } i ∈ I . Assume on the contrary that there exists f ∈ H such that

∑ i ∈ I ‖ Λ i f ‖ 2 > D Λ ‖ f ‖ 2 .

Then there is a finite set J ⊂ I so that

∑ i ∈ J ‖ Λ i f ‖ 2 > D Λ ‖ f ‖ 2 .

Since M is dense in H , there exists h ∈ M satisfying

∑ i ∈ J ‖ Λ i h ‖ 2 > D Λ ‖ h ‖ 2 ,

a contradiction to (3.1). Next we show that { Λ i } i ∈ I satisfies the opposite inequality of a K-g-frame. Let f ∈ H , then for any ε > 0 there is g ∈ M such that ‖ f − g ‖ < ε . Clearly, from (3.1) we can get that C Λ ‖ K ∗ f ‖ 2 ≤ D Λ ‖ f ‖ 2 . Now

∑ i ∈ I ‖ Λ i f ‖ 2 1 / 2 = ‖ { Λ i f } i ∈ I ‖ ≥ ‖ { Λ i g } i ∈ I ‖ − ‖ { Λ i ( f − g ) } i ∈ I ‖ ≥ C Λ ‖ K ∗ g ‖ − D Λ ‖ f − g ‖ = D Λ ‖ f − g ‖ + C Λ ‖ K ∗ g ‖ − 2 D Λ ‖ f − g ‖ ≥ C Λ ‖ K ∗ ( f − g ) ‖ + C Λ ‖ K ∗ g ‖ − 2 D Λ ε ≥ C Λ ‖ K ∗ ( f − g ) + K ∗ g ‖ − 2 D Λ ε = C Λ ‖ K ∗ f ‖ − 2 D Λ ε .

Therefore,

∑ i ∈ I ‖ Λ i f ‖ 2 ≥ C Λ ‖ K ∗ f ‖ 2 ,

as ε → 0 . Similarly, we can show that { Γ i } i ∈ I is a K-g-frame for H with respect to { K i } i ∈ I with frame bounds C Γ and D Γ .

Since { Λ i } i ∈ σ 1 ∪ { Γ i } i ∈ σ 2 is a K-g-frame for M with respect to { K i } i ∈ I with frame bounds C and D for any partition { σ 1 , σ 2 } of I , it follows that { Λ i } i ∈ σ 1 ∪ { Γ i } i ∈ σ 2 is a K-g-frame for H with respect to { K i } i ∈ I with frame bounds C and D . That is, { Λ i } i ∈ I and { Γ i } i ∈ I are K-woven in H with the universal frame bounds C and D .□

We now pay attention to constructing K-woven pair of K-g-frames for subspaces.

Theorem 3.5

Suppose that T , U ∈ B ( H ) has closed ranges, that { Λ i } i ∈ I and { Γ i } i ∈ I are K-g-frames for H with respect to { K i } i ∈ I , that Ker ( T ) = Ker ( U ) with K U T † = U T † K , and that Range ( K ∗ ) ⊂ Range ( U ) . If { Λ i T ∗ } i ∈ I and { Γ i T ∗ } i ∈ I are K-woven in Range ( T ) , then { Λ i U ∗ } i ∈ I and { Γ i U ∗ } i ∈ I are K-woven in Range ( U ) .

Proof

Define the operator L : Range ( T ) → Range ( U ) by L f = U T † f . Since Ker T = Ker U , we have Range ( T ∗ ) = ( Ker T ) ⊥ = ( Ker U ) ⊥ = Range ( U ∗ ) . Thus by Lemma 1.3,

(3.2) Ker U T † = Ker T T † = ( Range ( T ) ) ⊥ , T † T = P Range ( T ∗ ) = P Range ( U ∗ ) = U † U , L T = U T † T = U U † U = U .

From (3.2) we see that

Ker L = Ker U T † ∩ Range ( T ) = ( Range ( T ) ) ⊥ ∩ Range ( T ) = { 0 } ,

implying that L is invertible on Range ( T ) . Let C ′ and D ′ be frame bounds of { Λ i T ∗ } i ∈ I . Since for each f ∈ Range ( U ) , L ∗ f ∈ Range ( T ) , we have

∑ i ∈ I ‖ Λ i U ∗ f ‖ 2 = ∑ i ∈ I ‖ Λ i ( L T ) ∗ f ‖ 2 = ∑ i ∈ I ‖ Λ i T ∗ L ∗ f ‖ 2 ≥ C ′ ‖ K ∗ L ∗ f ‖ 2 = C ′ ‖ L ∗ K ∗ f ‖ 2 ≥ C ′ ‖ L − 1 ‖ − 2 ‖ K ∗ f ‖ 2 .

It is trivial to show that

∑ i ∈ I ‖ Λ i U ∗ f ‖ 2 = ∑ i ∈ I ‖ Λ i T ∗ L ∗ f ‖ 2 ≤ D ′ ‖ L ∗ f ‖ 2 ≤ D ′ ‖ L ‖ 2 ‖ f ‖ 2

for every f ∈ Range ( U ) . Hence, { Λ i U ∗ } i ∈ I is a K-g-frame for Range ( U ) with respect to { K i } i ∈ I . Similarly we can prove that { Γ i U ∗ } i ∈ I is a K-g-frame for Range ( U ) with respect to { K i } i ∈ I when { Γ i T ∗ } i ∈ I is a K-g-frame for Range ( T ) with respect to { K i } i ∈ I .

Since, for any partition { σ 1 , σ 2 } of I , { Λ i T ∗ } i ∈ σ 1 ∪ { Γ i T ∗ } i ∈ σ 2 is a K-g-frame for Range ( T ) with respect to { K i } i ∈ I , it follows that { Λ i U ∗ } i ∈ σ 1 ∪ { Γ i U ∗ } i ∈ σ 2 is a K-g-frame for Range ( U ) with respect to { K i } i ∈ I . That is, { Λ i U ∗ } i ∈ I and { Γ i U ∗ } i ∈ I are K-woven in Range ( U ) .□

The following result tells us that if one K-g-frame is K-woven to two K-g-frames, then it can be K-woven to their sum in Range ( K ) .

Theorem 3.6

Let K ∈ B ( H ) be a positive and closed range operator, and { Λ i } i ∈ I , { Γ i } i ∈ I and { Θ i } i ∈ I be K-g-frames for H with respect to { K i } i ∈ I . Assume that { Λ i } i ∈ I and { Γ i } i ∈ I , and { Λ i } i ∈ I and { Θ i } i ∈ I are both K-woven in H with the universal lower frame bounds C ′ and C ″ , respectively. Then { Λ i } i ∈ I and { Γ i + Θ i } i ∈ I are K-woven in Range ( K ) with the universal frame bounds ( C ′ + C ″ − ( D Λ + 2 D Γ D Θ ) ‖ K † ‖ 2 ) and D Λ + 2 ( D Γ + D Θ ) , if the following two conditions hold:

  1. K f = ∑ i ∈ I Γ i ∗ Θ i f for each f ∈ H ;

  2. C Γ > D Θ ‖ K † ‖ 2 and C ′ + C ″ > ( D Λ + 2 D Γ D Θ ) ‖ K † ‖ 2 .

Proof

We conclude first that { Γ i + Θ i } i ∈ I is a K-g-frame for Range ( K ) with respect to { K i } i ∈ I . As a matter of fact, for any f ∈ Range ( K ) we have

∑ i ∈ I ‖ ( Γ i + Θ i ) f ‖ 2 1 / 2 = ‖ { Γ i } i ∈ I + { Θ i } i ∈ I ‖ ≥ ‖ { Γ i } i ∈ I ‖ − ‖ { Θ i } i ∈ I ‖ ≥ C Γ ‖ K ∗ f ‖ − D Θ ‖ f ‖ = C Γ ‖ K ∗ f ‖ − D Θ ‖ K K † f ‖ = C Γ ‖ K ∗ f ‖ − D Θ ‖ ( K † ) ∗ K ∗ f ‖ ≥ ( C Γ − D Θ ‖ K † ‖ ) ‖ K ∗ f ‖ .

The g-Bessel bound of { Γ i + Θ i } i ∈ I is trivial. Next we show that { Λ i } i ∈ I and { Γ i + Θ i } i ∈ I are K-woven in Range ( K ) . For any partition { σ 1 , σ 2 } of I , let

(3.3) T Γ σ 1 ( { g i } i ∈ I ) = ∑ i ∈ σ 1 Γ i ∗ g i and T Θ σ 2 ( { g i } i ∈ I ) = ∑ i ∈ σ 2 Θ i ∗ g i

for any { g i } i ∈ I ∈ ℓ 2 ( { K i } j ∈ I ) . Then it is easy to check that ‖ T Γ σ 1 ‖ ≤ ‖ T Γ ‖ ≤ D Γ and ‖ T Θ σ 2 ‖ ≤ ‖ T Θ ‖ ≤ D Θ . Now for any f ∈ Range ( K ) we compute

∑ i ∈ σ 1 ‖ Λ i f ‖ 2 + ∑ i ∈ σ 2 ‖ ( Γ i + Θ i ) f ‖ 2 = ∑ i ∈ σ 1 ‖ Λ i f ‖ 2 + ∑ i ∈ σ 2 ‖ Γ i f ‖ 2 + ∑ i ∈ σ 2 ‖ Θ i f ‖ 2 + ∑ i ∈ σ 2 ⟨ Γ i ∗ Θ i f , f ⟩ + ∑ i ∈ σ 2 ⟨ Θ i ∗ Γ i f , f ⟩ = ∑ i ∈ σ 1 ‖ Λ i f ‖ 2 + ∑ i ∈ σ 2 ‖ Γ i f ‖ 2 + ∑ i ∈ σ 1 ‖ Λ i f ‖ 2 + ∑ i ∈ σ 2 ‖ Θ i f ‖ 2 + ⟨ K f , f ⟩ + ⟨ K ∗ f , f ⟩ − ∑ i ∈ σ 1 ⟨ Γ i ∗ Θ i f , f ⟩ + ∑ i ∈ σ 1 ⟨ Θ i ∗ Γ i f , f ⟩ − ∑ i ∈ σ 1 ‖ Λ i f ‖ 2 ≥ C ′ ‖ K ∗ f ‖ 2 + C ″ ‖ K ∗ f ‖ 2 − D Λ ‖ f ‖ 2 − ∑ i ∈ σ 1 ⟨ Γ i ∗ Θ i f , f ⟩ + ∑ i ∈ σ 1 ⟨ Θ i ∗ Γ i f , f ⟩ ≥ ( C ′ + C ″ ) ‖ K ∗ f ‖ 2 − D Λ ‖ f ‖ 2 − ‖ f ‖ ‖ T Γ σ 1 T Θ ∗ f + T Θ σ 1 T Γ ∗ f ‖ ≥ ( C ′ + C ″ ) ‖ K ∗ f ‖ 2 − ( D Λ + 2 D Γ D Θ ) ‖ f ‖ 2 ≥ ( C ′ + C ″ ) ‖ K ∗ f ‖ 2 − ( D Λ + 2 D Γ D Θ ) ‖ K † ‖ 2 ‖ K ∗ f ‖ 2 = ( C ′ + C ″ − ( D Λ + 2 D Γ D Θ ) ‖ K † ‖ 2 ) ‖ K ∗ f ‖ 2 .

The fact that D Λ + 2 ( D Γ + D Θ ) is an upper weaving bound is obvious.□

We conclude this section with the result showing that we can use operator K and the corresponding synthesis operators to construct K-woven pair of K-g-frames.

Theorem 3.7

Let { Λ i } i ∈ I and { Γ i } i ∈ I be two K-g-frames for H with respect to { K i } i ∈ I . For any partition { σ 1 , σ 2 } of I , let T and S be, respectively, the synthesis operator and g-frame operator of { Λ i } i ∈ σ 1 ∪ { Γ i } i ∈ σ 2 . Then { Λ i } i ∈ I and { Γ i } i ∈ I are K-woven in H with the universal frame bounds C and D , if the following two conditions are satisfied:

  1. Range ( K ) ⊂ Range ( T ) and Range ( S ) ⊂ Range ( T ) ;

  2. There exist two constants C , D > 0 such that

    (3.4) C ‖ K ‖ 2 ‖ { g i } i ∈ I ‖ 2 ≤ ‖ T { g i } i ∈ I ‖ 2 ≤ D ‖ { g i } i ∈ I ‖ 2 , ∀ { g i } i ∈ I ∈ Ker ⊥ ( T ) .

Proof

We first prove that T has closed range. Suppose that { h n } ⊂ Range ( T ) and h n → h as n → ∞ . Since ℓ 2 ( { K i } i ∈ I ) = Ker ( T ) ⊕ Ker ⊥ ( T ) , we can find { x n } ⊂ Ker ⊥ ( T ) such that h n = T x n . For any m , n ∈ N , we see from (3.4) that

‖ x n − x m ‖ 2 ≤ C − 1 ‖ K ‖ − 2 ‖ T x n − T x m ‖ 2 = C − 1 ‖ K ‖ − 2 ‖ h n − h m ‖ 2 ,

it follows that { x n } is a Cauchy sequence. Assume that x n → x ∈ ℓ 2 ( { K i } i ∈ I ) , then by the continuity of T we have h n = T x n → T x = h ∈ Range ( T ) as n → ∞ . Thus, Range ( T ) is closed. For each f ∈ H , there exist g ∈ Range ( T ) and h ∈ ( Range ( T ) ) ⊥ = Ker ( T † ) such that f = g + h . Then T † f = T † g ∈ Range ( T † ) = Ker ⊥ ( T ) . It follows, by combining Lemma 1.3 and (3.4), that

‖ T † f ‖ 2 = ‖ T † g ‖ 2 ≤ C − 1 ‖ K ‖ − 2 ‖ T T † g ‖ 2 = C − 1 ‖ K ‖ − 2 ‖ g ‖ 2 ≤ C − 1 ‖ K ‖ − 2 ( ‖ g ‖ 2 + ‖ h ‖ 2 ) = C − 1 ‖ K ‖ − 2 ‖ f ‖ 2 .

Hence, ‖ T † ‖ 2 ≤ C − 1 ‖ K ‖ − 2 . Now for any g ∈ Range ( T ) we have

‖ K ∗ g ‖ 2 ≤ ‖ K ‖ 2 ‖ g ‖ 2 = ‖ K ‖ 2 ‖ T T † g ‖ 2 = ‖ K ‖ 2 ‖ ( T † ) ∗ T ∗ g ‖ 2 ≤ ‖ K ‖ 2 ‖ T † ‖ 2 ‖ T ∗ g ‖ 2 ≤ C − 1 ‖ T ∗ g ‖ 2 = C − 1 ∑ i ∈ σ 1 ‖ Λ i g ‖ 2 + ∑ i ∈ σ 2 ‖ Γ i g ‖ 2 .

Therefore,

C ‖ K ∗ g ‖ 2 ≤ ∑ i ∈ σ 1 ‖ Λ i g ‖ 2 + ∑ i ∈ σ 2 ‖ Γ i g ‖ 2 .

For each f ∈ H we have f = g + h with g ∈ Range ( T ) and h ∈ ( Range ( T ) ) ⊥ . Since Range ( K ) ⊂ Range ( T ) and Range ( S ) ⊂ Range ( T ) , it follows that

∑ i ∈ σ 1 ‖ Λ i f ‖ 2 + ∑ i ∈ σ 2 ‖ Γ i f ‖ 2 = ∑ i ∈ σ 1 ‖ Λ i ( g + h ) ‖ 2 + ∑ i ∈ σ 2 ‖ Γ i ( g + h ) ‖ 2 = ∑ i ∈ σ 1 ‖ Λ i g ‖ 2 + ∑ i ∈ σ 1 Λ i ∗ Λ i g , h + h , ∑ i ∈ σ 1 Λ i ∗ Λ i g + ∑ i ∈ σ 1 Λ i ∗ Λ i h , h + ∑ i ∈ σ 2 ‖ Γ i g ‖ 2 + ∑ i ∈ σ 2 Γ i ∗ Γ i g , h + h , ∑ i ∈ σ 2 Γ i ∗ Γ i g + ∑ i ∈ σ 2 Γ i ∗ Γ i h , h = ∑ i ∈ σ 1 ‖ Λ i g ‖ 2 + ∑ i ∈ σ 2 ‖ Γ i g ‖ 2 + ⟨ S g , h ⟩ + ⟨ h , S g ⟩ + ⟨ S h , h ⟩ = ∑ i ∈ σ 1 ‖ Λ i g ‖ 2 + ∑ i ∈ σ 2 ‖ Γ i g ‖ 2 ≥ C ‖ K ∗ g ‖ 2 = C ‖ K ∗ ( g + h ) ‖ 2 = C ‖ K ∗ f ‖ 2 .

Next we prove the opposite inequality. For any g ∈ ℓ 2 ( { K i } i ∈ I ) , there exist g 1 ∈ Ker ( T ) and g 2 ∈ Ker ⊥ ( T ) such that g = g 1 + g 2 . Then the right-hand side of inequality of (3.4) leads to

‖ T g ‖ 2 = ‖ T g 2 ‖ 2 ≤ D ‖ g 2 ‖ 2 ≤ D ( ‖ g 1 ‖ 2 + ‖ g 2 ‖ 2 ) = D ‖ g ‖ 2 .

Thus ‖ T ‖ ≤ D and, consequently,

∑ i ∈ σ 1 ‖ Λ i f ‖ 2 + ∑ i ∈ σ 2 ‖ Γ i f ‖ 2 = ‖ T ∗ f ‖ 2 ≤ D ‖ f ‖ 2 , ∀ f ∈ H .

Altogether we obtain

C ‖ K ∗ f ‖ 2 ≤ ∑ i ∈ σ 1 ‖ Λ i f ‖ 2 + ∑ i ∈ σ 2 ‖ Γ i f ‖ 2 ≤ D ‖ f ‖ 2 , ∀ f ∈ H .

This completes the proof.□

4 A new Paley-Wiener-type perturbation result on the weaving of K-g-frames

The perturbation theory for frames is important in practice and therefore many researchers were interested and some significant results were presented, see, for example, the papers [26,28]. Recently, the authors in [25] proved a Paley-Wiener-type perturbation theorem (Theorem 3) for the weaving of K-g-frames in a subspace of H . We show in the following result, however, that under the perturbation condition involved in that theorem, two K-g-frames can be K-woven in H .

Theorem 4.1

Let { Λ i } i ∈ I and { Γ i } i ∈ I be K-g-frames for H with respect to { K i } i ∈ I . Assume that Range ( T Λ σ 1 ) ⊂ Range ( K ) and Range ( T Γ σ 2 ) ⊂ Range ( K ) for any partition { σ 1 , σ 2 } of I , where T Λ σ 1 and T Γ σ 2 are defined the same way as in (3.3). If there are constants α , β , γ ≥ 0 such that C Λ − ( D Λ + D Γ ) ( α D Λ + β D Γ + γ ) ‖ K ∗ ˜ − 1 ‖ 2 > 0 and

(4.1) ∑ i ∈ I ( Γ i ∗ − Λ i ∗ ) g i ≤ α ∑ i ∈ I Λ i ∗ g i + β ∑ i ∈ I Γ i ∗ g i + γ ‖ { g i } i ∈ I ‖

for any { g i } i ∈ I ∈ ℓ 2 ( { K i } i ∈ I ) , then { Λ i } i ∈ I and { Γ i } i ∈ I are K-woven in H with the universal frame bounds C Λ − ( D Λ + D Γ ) ( α D Λ + β D Γ + γ ) ‖ K ∗ ˜ − 1 ‖ 2 and D Λ + D Γ , where K ∗ ˜ : Ker ⊥ ( K ∗ ) → Range ( K ∗ ) is the restriction of K ∗ on Ker ⊥ ( K ∗ ) .

Proof

It is easy to check that { Λ i } i ∈ I and { Γ i } i ∈ I always admit an universal upper frame bound D Λ + D Γ . So it only needs to prove the universal lower frame bound condition. Since ‖ T Λ ‖ ≤ D Λ and ‖ T Γ ‖ ≤ D Γ , by using (4.1) we compute that

‖ T Γ { g i } i ∈ I − T Λ { g i } i ∈ I ‖ = ∑ i ∈ I ( Γ i ∗ − Λ i ∗ ) g i ≤ α ∑ i ∈ I Λ i ∗ g i + β ∑ i ∈ I Γ i ∗ g i + γ ‖ { g i } i ∈ I ‖ = α ‖ T Λ { g i } i ∈ I ‖ + β ‖ T Γ { g i } i ∈ I ‖ + γ ‖ { g i } i ∈ I ‖ ≤ α D Λ ‖ { g i } i ∈ I ‖ + β D Γ ‖ { g i } i ∈ I ‖ + γ ‖ { g i } i ∈ I ‖ = ( α D Λ + β D Γ + γ ) ‖ { g i } i ∈ I ‖

for any { g i } i ∈ I ∈ ℓ 2 ( { K i } i ∈ I ) , which leads to ‖ T Γ − T Λ ‖ ≤ α D Λ + β D Γ + γ . Hence, for any partition { σ 1 , σ 2 } of I we get

‖ T Λ σ 2 T Λ ∗ − T Γ σ 2 T Γ ∗ ‖ = ‖ ( T Λ σ 2 T Λ ∗ − T Λ σ 2 T Γ ∗ ) + ( T Λ σ 2 T Γ ∗ − T Γ σ 2 T Γ ∗ ) ‖ ≤ ‖ T Λ σ 2 T Λ ∗ − T Λ σ 2 T Γ ∗ ‖ + ‖ T Λ σ 2 T Γ ∗ − T Γ σ 2 T Γ ∗ ‖ ≤ ‖ T Λ σ 2 ‖ ‖ T Λ ∗ − T Γ ∗ ‖ + ‖ T Λ σ 2 − T Γ σ 2 ‖ ‖ T Γ ∗ ‖ ≤ ( ‖ T Λ ‖ + ‖ T Γ ‖ ) ‖ T Λ − T Γ ‖ ≤ ( D Λ + D Γ ) ( α D Λ + β D Γ + γ ) ,

where in the third inequality we apply the fact that ‖ T Λ σ 2 − T Γ σ 2 ‖ ≤ ‖ T Λ − T Γ ‖ . Thus,

∑ i ∈ σ 1 ‖ Λ i g ‖ 2 + ∑ i ∈ σ 2 ‖ Γ i g ‖ 2 = ∑ i ∈ σ 1 ‖ Λ i g ‖ 2 + ∑ i ∈ σ 2 ‖ Λ i g ‖ 2 − ∑ i ∈ σ 2 ‖ Λ i g ‖ 2 − ∑ i ∈ σ 2 ‖ Γ i g ‖ 2 = ∑ i ∈ I ‖ Λ i g ‖ 2 − ∑ i ∈ σ 2 ( Λ i ∗ Λ i g − Γ i ∗ Γ i g ) , g

≥ C Λ ‖ K ∗ g ‖ 2 − ‖ g ‖ ∑ i ∈ σ 2 Λ i ∗ Λ i g − ∑ i ∈ σ 2 Γ i ∗ Γ i g = C Λ ‖ K ∗ g ‖ 2 − ‖ g ‖ ‖ T Λ σ 2 T Λ ∗ g − T Γ σ 2 T Γ ∗ g ‖ ≥ C Λ ‖ K ∗ g ‖ 2 − ( D Λ + D Γ ) ( α D Λ + β D Γ + γ ) ‖ g ‖ 2 = C Λ ‖ K ∗ g ‖ 2 − ( D Λ + D Γ ) ( α D Λ + β D Γ + γ ) ‖ K ∗ ˜ − 1 K ∗ ˜ g ‖ 2 ≥ ( C Λ − ( D Λ + D Γ ) ( α D Λ + β D Γ + γ ) ‖ K ∗ ˜ − 1 ‖ 2 ) ‖ K ∗ g ‖ 2

for any g ∈ Ker ⊥ ( K ∗ ) . Noting that each f ∈ H has a decomposition f = f 1 + f 2 with f 1 ∈ Ker ( K ∗ ) and f 2 ∈ Ker ⊥ ( K ∗ ) , and that Range ( T Λ σ 1 ) ⊂ Range ( K ) and Range ( T Γ σ 2 ) ⊂ Range ( K ) , we obtain

∑ i ∈ σ 1 ‖ Λ i f ‖ 2 + ∑ i ∈ σ 2 ‖ Γ i f ‖ 2 = ∑ i ∈ σ 1 ‖ Λ i ( f 1 + f 2 ) ‖ 2 + ∑ i ∈ σ 2 ‖ Γ i ( f 1 + f 2 ) ‖ 2 = ∑ i ∈ σ 1 ‖ Λ i f 2 ‖ 2 + ∑ i ∈ σ 1 Λ i ∗ Λ i f 1 , f 1 + 2 Re ∑ i ∈ σ 1 Λ i ∗ Λ i f 2 , f 1 + ∑ i ∈ σ 2 ‖ Γ i f 2 ‖ 2 + ∑ i ∈ σ 2 Γ i ∗ Γ i f 1 , f 1 + 2 Re ∑ i ∈ σ 2 Γ i ∗ Γ i f 2 , f 1 = ∑ i ∈ σ 1 ‖ Λ i f 2 ‖ 2 + ∑ i ∈ σ 2 ‖ Γ i f 2 ‖ 2 + ⟨ T Λ σ 1 T Λ ∗ f 1 , f 1 ⟩ + 2 Re ⟨ T Λ σ 1 T Λ ∗ f 2 , f 1 ⟩ + ⟨ T Γ σ 2 T Γ ∗ f 1 , f 1 ⟩ + 2 Re ⟨ T Γ σ 2 T Γ ∗ f 2 , f 1 ⟩ = ∑ i ∈ σ 1 ‖ Λ i f 2 ‖ 2 + ∑ i ∈ σ 2 ‖ Γ i f 2 ‖ 2 ≥ ( C Λ − ( D Λ + D Γ ) ( α D Λ + β D Γ + γ ) ‖ K ∗ ˜ − 1 ‖ 2 ) ‖ K ∗ f 2 ‖ 2 = ( C Λ − ( D Λ + D Γ ) ( α D Λ + β D Γ + γ ) ‖ K ∗ ˜ − 1 ‖ 2 ) ‖ K ∗ ( f 1 + f 2 ) ‖ 2 = ( C Λ − ( D Λ + D Γ ) ( α D Λ + β D Γ + γ ) ‖ K ∗ ˜ − 1 ‖ 2 ) ‖ K ∗ f ‖ 2 ,

and we have the result.□

Acknowledgments

The author thanks the anonymous reviewers and the editor for their valuable suggestions and comments which have led to a significant improvement of this manuscript.

  1. Funding information: The research was supported by the National Natural Science Foundation of China (Nos 11761057 and 11561057) and the Science Foundation of Jiangxi Education Department (Nos GJJ202302 and GJJ190886).

  2. Author contributions: The author has accepted responsibility for the entire content of this manuscript and approved its submission.

  3. Conflict of interest: The author states no conflict of interest.

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Received: 2020-11-06
Revised: 2021-09-11
Accepted: 2021-09-12
Published Online: 2021-12-31

© 2021 Zhong-Qi Xiang, published by De Gruyter

This work is licensed under the Creative Commons Attribution 4.0 International License.

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https://doi.org/10.1515/math-2021-0103
Keywords for this article
K-g-frame; K-woven; perturbation; pseudo-inverse
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  79. A posteriori error estimates based on superconvergence of FEM for fractional evolution equations
  80. Solution of integral equations via coupled fixed point theorems in 𝔉-complete metric spaces
  81. Symmetric pairs and pseudosymmetry of Θ-Yetter-Drinfeld categories for Hom-Hopf algebras
  82. A new characterization of the automorphism groups of Mathieu groups
  83. The role of w-tilting modules in relative Gorenstein (co)homology
  84. Primitive and decomposable elements in homology of ΩΣℂP∞
  85. The G-sequence shadowing property and G-equicontinuity of the inverse limit spaces under group action
  86. Classification of f-biharmonic submanifolds in Lorentz space forms
  87. Some new results on the weaving of K-g-frames in Hilbert spaces
  88. Matrix representation of a cross product and related curl-based differential operators in all space dimensions
  89. Global optimization and applications to a variational inequality problem
  90. Functional equations related to higher derivations in semiprime rings
  91. A partial order on transformation semigroups with restricted range that preserve double direction equivalence
  92. On multi-step methods for singular fractional q-integro-differential equations
  93. Compact perturbations of operators with property (t)
  94. Entire solutions for several complex partial differential-difference equations of Fermat type in ℂ2
  95. Random attractors for stochastic plate equations with memory in unbounded domains
  96. On the convergence of two-step modulus-based matrix splitting iteration method
  97. On the separation method in stochastic reconstruction problem
  98. Robust estimation for partial functional linear regression models based on FPCA and weighted composite quantile regression
  99. Structure of coincidence isometry groups
  100. Sharp function estimates and boundedness for Toeplitz-type operators associated with general fractional integral operators
  101. Oscillatory hyper-Hilbert transform on Wiener amalgam spaces
  102. Euler-type sums involving multiple harmonic sums and binomial coefficients
  103. Poly-falling factorial sequences and poly-rising factorial sequences
  104. Geometric approximations to transition densities of Jump-type Markov processes
  105. Multiple solutions for a quasilinear Choquard equation with critical nonlinearity
  106. Bifurcations and exact traveling wave solutions for the regularized Schamel equation
  107. Almost factorizable weakly type B semigroups
  108. The finite spectrum of Sturm-Liouville problems with n transmission conditions and quadratic eigenparameter-dependent boundary conditions
  109. Ground state sign-changing solutions for a class of quasilinear Schrödinger equations
  110. Epi-quasi normality
  111. Derivative and higher-order Cauchy integral formula of matrix functions
  112. Commutators of multilinear strongly singular integrals on nonhomogeneous metric measure spaces
  113. Solutions to a multi-phase model of sea ice growth
  114. Existence and simulation of positive solutions for m-point fractional differential equations with derivative terms
  115. Bernstein-Walsh type inequalities for derivatives of algebraic polynomials in quasidisks
  116. Review Article
  117. Semiprimeness of semigroup algebras
  118. Special Issue on Problems, Methods and Applications of Nonlinear Analysis (Part II)
  119. Third-order differential equations with three-point boundary conditions
  120. Fractional calculus, zeta functions and Shannon entropy
  121. Uniqueness of positive solutions for boundary value problems associated with indefinite ϕ-Laplacian-type equations
  122. Synchronization of Caputo fractional neural networks with bounded time variable delays
  123. On quasilinear elliptic problems with finite or infinite potential wells
  124. Deterministic and random approximation by the combination of algebraic polynomials and trigonometric polynomials
  125. On a fractional Schrödinger-Poisson system with strong singularity
  126. Parabolic inequalities in Orlicz spaces with data in L1
  127. Special Issue on Evolution Equations, Theory and Applications (Part II)
  128. Impulsive Caputo-Fabrizio fractional differential equations in b-metric spaces
  129. Existence of a solution of Hilfer fractional hybrid problems via new Krasnoselskii-type fixed point theorems
  130. On a nonlinear system of Riemann-Liouville fractional differential equations with semi-coupled integro-multipoint boundary conditions
  131. Blow-up results of the positive solution for a class of degenerate parabolic equations
  132. Long time decay for 3D Navier-Stokes equations in Fourier-Lei-Lin spaces
  133. On the extinction problem for a p-Laplacian equation with a nonlinear gradient source
  134. General decay rate for a viscoelastic wave equation with distributed delay and Balakrishnan-Taylor damping
  135. On hyponormality on a weighted annulus
  136. Exponential stability of Timoshenko system in thermoelasticity of second sound with a memory and distributed delay term
  137. Convergence results on Picard-Krasnoselskii hybrid iterative process in CAT(0) spaces
  138. Special Issue on Boundary Value Problems and their Applications on Biosciences and Engineering (Part I)
  139. Marangoni convection in layers of water-based nanofluids under the effect of rotation
  140. A transient analysis to the M(τ)/M(τ)/k queue with time-dependent parameters
  141. Existence of random attractors and the upper semicontinuity for small random perturbations of 2D Navier-Stokes equations with linear damping
  142. Degenerate binomial and Poisson random variables associated with degenerate Lah-Bell polynomials
  143. Special Issue on Fractional Problems with Variable-Order or Variable Exponents (Part I)
  144. On the mixed fractional quantum and Hadamard derivatives for impulsive boundary value problems
  145. The Lp dual Minkowski problem about 0 < p < 1 and q > 0
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