New forms of bilateral inequalities for K-g-frames

Zhong-Qi Xiang

doi:10.1515/dema-2025-0110

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New forms of bilateral inequalities for K-g-frames

Published/Copyright: March 19, 2025

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From the journal Demonstratio Mathematica Volume 58 Issue 1

Abstract

In this work, several bilateral inequalities for K - g -frames in subspaces are established, drawing support from two kinds of operators induced, respectively, by the K - g -frame itself and the K -dual pair, which, compared with previous ones concerning this topic, possess novel structures. It is indicated that new types of inequalities for some other generalized frames can be naturally presented following our approaches.

Keywords: K-dual pair; pseudo-inverse operator; bilateral inequality

MSC 2010: 42C15; 42C40; 46C50; 47B40

1 Introduction and preliminaries

Throughout this article, I and R are, respectively, a countable index set and the set of real numbers, ℳ , P , and N i ( i ∈ I ) are (complex separable) Hilbert spaces. For any σ ⊂ I , we let σ c be the complementary set of σ . The notation ℒℬ ( ℳ , P ) is designated as the family of linear bounded operators from ℳ to P , and ℒℬ ( ℳ , ℳ ) is simply denoted by ℒℬ ( ℳ ) . For K ∈ ℒℬ ( ℳ ) , we denote, respectively, by Ran ( K ) and Ker ( K ) the range and the null space of K . We use, as usual, the symbol Id ℳ to denote the identity operator on ℳ .

Frames (embedded in Hilbert spaces), also known as redundant bases, were proposed by Duffin and Schaeffer [1] on account of an in-depth research on nonharmonic Fourier series in 1952. More than 30 years later, Daubechies et al. [2] brought frames back to people’s vision owing to their groundbreaking work on wavelets. Thanks to some of their nice properties, frames have already been applied to many research fields (see [3–6]). We refer also to [7] for more details about frame theory.

In the past two decades, the notion of frames was rapidly extended, and a number of generalized frame forms have emerged, among which, g -frames introduced by Sun [8] unify some other generalizations of frames and admit some properties that are distinct from those for frames [9]. Drawing on the idea of [10], Zhou and Zhu in [11] brought forward a more general frame concept, namely, K - g -frames, by combining the linear bounded operator K and g -frames, which behave, in some ways, quite distinct from g -frames (See also [12]).

Definition 1.1

(See [11, Definition 1.3]) Let K ∈ ℒℬ ( ℳ ) , and assume that Λ i ∈ ℒℬ ( ℳ , N i ) for each i ∈ I . One calls the family { Λ i } i ∈ I a K-g-frame for ℳ with respect to { N i } i ∈ I , if there exist numbers 0 < C Λ ≤ D Λ < ∞ , called the K-g-frame bounds of { Λ i } i ∈ I , so that the two-sided inequality

(1.1) C Λ ‖ K * x ‖ 2 ≤ ∑ i ∈ I ‖ Λ i x ‖ 2 ≤ D Λ ‖ x ‖ 2

holds for every x ∈ ℳ . The K - g -frame { Λ i } i ∈ I is said to be Parseval if

(1.2) ‖ K * x ‖ 2 = ∑ i ∈ I ‖ Λ i x ‖ 2 , ∀ x ∈ ℳ .

Remark 1.2

In the sequel, we just say, for example, that { Λ i } i ∈ I is a K - g -frame for ℳ without emphasizing the space sequence { N i } i ∈ I , since it is clear.

If only the inequality on the right-hand side of (1.1) is assumed to be satisfied, then { Λ i } i ∈ I is said to be a g-Bessel sequence for ℳ , which can naturally lead to a self-adjoint and positive operator, called the g-frame operator of { Λ i } i ∈ I , defined by

(1.3) S Λ : ℳ → ℳ , S Λ x = ∑ i ∈ I Λ i ∗ Λ i x .

If we let K = Id ℳ in Definition 1.1, then the K - g -frame { Λ i } i ∈ I turns to be a g -frame, and, in this setting, the g -frame operator S Λ is invertible (see [8]).

Definition 1.3

(See [13, Definition 1.3]) Let { Λ i } i ∈ I be a K - g -frame for ℳ . We call a g -Bessel sequence { Γ i } i ∈ I for ℳ a dual K-g-Bessel sequence of { Λ i } i ∈ I , if

(1.4) K x = ∑ i ∈ I Λ i ∗ Γ i x , ∀ x ∈ ℳ .

In this case, the pair ( { Λ i } i ∈ I , { Γ i } i ∈ I ) is said to be a K-dual pair.

If Θ ∈ ℒℬ ( ℳ , P ) admits closed range, then there is a unique operator Θ † ∈ ℒℬ ( P , ℳ ) , namely the pseudo-inverse operator, possessing the following properties:

(1.5) Θ Θ † Θ = Θ , Θ † Θ Θ † = Θ † , Ker ( Θ † ) = ( Ran ( Θ ) ) ⊥ , Ran ( Θ † ) = ( Ker ( Θ ) ) ⊥ .

See [7] for more information.

Suppose that K ∈ ℒℬ ( ℳ ) has closed range and that { Λ i } i ∈ I is a K - g -frame for ℳ with g -frame bounds C Λ and D Λ . Since each x ∈ Ran ( K ) can be represented as x = K K † x , we obtain K K † ∣ Ran ( K ) = Id Ran ( K ) . Thus,

( K † ∣ Ran ( K ) ) * K * ∣ Ran ( K ) = Id Ran ( K ) ,

and further,

‖ x ‖ 2 = ‖ ( K † ∣ Ran ( K ) ) * K * x ‖ 2 ≤ ‖ K † ∣ Ran ( K ) ‖ 2 ‖ K * x ‖ 2

for every x ∈ Ran ( K ) . Therefore,

∑ i ∈ I ‖ ( Λ i ∣ Ran ( K ) ) x ‖ 2 = ∑ i ∈ I ‖ Λ i x ‖ 2 ≥ C Λ ‖ K * x ‖ 2 ≥ C Λ ‖ K † ∣ Ran ( K ) ‖ − 2 ‖ x ‖ 2 ,

implying that { Λ i ∣ Ran ( K ) } i ∈ I is a g -frame for Ran ( K ) .

It is easy to check that the operators S Λ σ , S Λ σ c : Ran ( K ) → Ran ( K ) given by

(1.6) S Λ σ x = ∑ i ∈ σ ( Λ i ∣ Ran ( K ) ) * ( Λ i ∣ Ran ( K ) ) x , S Λ σ c x = ∑ i ∈ σ c ( Λ i ∣ Ran ( K ) ) * ( Λ i ∣ Ran ( K ) ) x ,

are well-defined and bounded. If we denote by S Λ ′ the g -frame operator of { Λ i ∣ Ran ( K ) } i ∈ I , then it is clear that

(1.7) S Λ ′ = S Λ σ + S Λ σ c .

As the derivate of the famous Parseval frame identity arising in the process of exploring efficient algorithms for the reconstruction of signals [14], Balan et al. [15] offered us an interesting inequality for Parseval frames, which is listed as follows.

Theorem A

(See [15, Proposition 4.1]) Suppose that { ψ i } i ∈ I is a Parseval frame for ℳ . Then, for every x ∈ ℳ and any σ ⊂ I , we obtain

(1.8) ∑ i ∈ σ ∣ ⟨ x , ψ i ⟩ ∣ 2 + ∑ i ∈ σ c ⟨ x , ψ i ⟩ ψ i 2 ≥ 3 4 ‖ x ‖ 2 .

Later, Găvruţa [16] extended inequality (1.8) to the case of general frames, assisted by the corresponding canonical dual frames and alternate dual frames, given below.

Theorem B

(See [16, Theorem 2.2]) Assume that { ψ i } i ∈ I is a frame for ℳ and that { ψ ˜ i } i ∈ I is the canonical dual frame of { ψ i } i ∈ I . Then, for every x ∈ ℳ and any σ ⊂ I , we have

(1.9) ∑ i ∈ σ ∣ ⟨ x , ψ i ⟩ ∣ 2 + ∑ i ∈ I ∣ ⟨ S σ c x , ψ ˜ i ⟩ ∣ 2 ≥ 3 4 ∑ i ∈ I ∣ ⟨ x , ψ i ⟩ ∣ 2 .

Theorem C

(See [16, Theorem 3.2]) Assume that { ψ i } i ∈ I is a frame for ℳ and that { φ i } i ∈ I is an alternate dual frame of { ψ i } i ∈ I . Then, for every x ∈ ℳ and any σ ⊂ I , we have

(1.10) Re ∑ i ∈ σ ⟨ x , φ i ⟩ ⟨ ψ i , x ⟩ + ∑ i ∈ σ c ⟨ x , φ i ⟩ ψ i 2 ≥ 3 4 ‖ x ‖ 2 .

In recent years, much attention has been paid to the extension of the frame inequalities (1.8), (1.9), and (1.10), and many forms of inequalities for generalized frames were presented, see, e.g., the papers [17–24], which further enrich the inequality theory of frames. As for K - g -frames, however, it is difficult to set up inequalities like other generalized versions of frames, due to the operator K . Fortunately, by means of the pseudo-inverse operator of K , Xiao et al. [25] provided us some inequalities for K - g -frames in subspaces.

Theorem D

(See [25, Theorem 1]) Let K ∈ ℒℬ ( ℳ ) and assume that { Λ i } i ∈ I is a K-g-frame for ℳ , and that { Γ i } i ∈ I is a dual K-g-Bessel sequence of { Λ i } i ∈ I . If Ran ( K ) is closed, then the following statements hold.

For each x ∈ Ran ( K ) , for any λ ∈ [ 0,1 ] and any a = { a i } i ∈ I ∈ ℓ ∞ ( I ) , we have
∑ i ∈ I a i Λ i * Γ i K † x 2 + 2 λ Re ∑ i ∈ I ( 1 − a i ) ⟨ Γ i K † x , Λ i x ⟩ ≥ ( 2 λ − λ 2 ) ‖ x ‖ 2 .
For each x ∈ Ran ( K ) , for any λ ∈ R and any a = { a i } i ∈ I ∈ ℓ ∞ ( I ) , we have
∑ i ∈ I a i Λ i * Γ i K † x 2 + λ Re ∑ i ∈ I a i ⟨ Γ i K † x , Λ i x ⟩ + Re ∑ i ∈ I ( 1 − a ¯ i ) ⟨ Λ i x , Γ i K † x ⟩ ≥ 3 4 − 1 4 λ 2 + λ 2 ‖ x ‖ 2 .

Remark 1.4

As a matter of fact, the inequality in (1) holds for every λ ∈ R . Indeed, for each x ∈ Ran ( K ) and each λ ∈ R , we have

∑ i ∈ I a i Λ i * Γ i K † x − λ x 2 + ( 2 λ − λ 2 ) ‖ x ‖ 2 ≥ ( 2 λ − λ 2 ) ‖ x ‖ 2 .

That is

(1.11) ∑ i ∈ I a i Λ i * Γ i K † x 2 + λ 2 ‖ x ‖ 2 − 2 λ Re ∑ i ∈ I a i Λ i * Γ i K † x , x + ( 2 λ − λ 2 ) ‖ x ‖ 2 ≥ ( 2 λ − λ 2 ) ‖ x ‖ 2 .

Noting also that

Re ∑ i ∈ I ( 1 − a i ) ⟨ Γ i K † x , Λ i x ⟩ = Re ∑ i ∈ I ( 1 − a i ) Λ i * Γ i K † x , x = Re ∑ i ∈ I Λ i * Γ i K † x , x − Re ∑ i ∈ I a i Λ i * Γ i K † x , x = ⟨ K K † x , x ⟩ − Re ∑ i ∈ I a i Λ i * Γ i K † x , x = ‖ x ‖ 2 − Re ∑ i ∈ I ⟨ a i Γ i K † x , Λ i x ⟩ .

This together with (1.11) yields

∑ i ∈ I a i Λ i * Γ i K † x 2 + 2 λ Re ∑ i ∈ I ( 1 − a i ) ⟨ Γ i K † x , Λ i x ⟩ ≥ ( 2 λ − λ 2 ) ‖ x ‖ 2 .

Theorem E

(See [25, Theorem 2]) Let K ∈ ℒℬ ( ℳ ) , and assume that { Λ i } i ∈ I is a K-g-frame for ℳ , and that { Γ i } i ∈ I is a dual K-g-Bessel sequence of { Λ i } i ∈ I . If Ran ( K ) is closed, then the following assertions hold.

For each x ∈ Ran ( K † ) , for any λ ∈ [ 0 , 1 ] and any a = { a i } i ∈ I ∈ ℓ ∞ ( I ) , we have
∑ i ∈ I a i K † Λ i * Γ i x 2 + 2 λ Re ∑ i ∈ I ( 1 − a i ) ⟨ Γ i x , Λ i ( K † ) * x ⟩ ≥ ( 2 λ − λ 2 ) ‖ x ‖ 2 .
For each x ∈ Ran ( K † ) , for any λ ∈ R and any a = { a i } i ∈ I ∈ ℓ ∞ ( I ) , we have
∑ i ∈ I a i K † Λ i * Γ i x 2 + λ Re ∑ i ∈ I a i ⟨ Γ i x , Λ i ( K † ) * x ⟩ + Re ∑ i ∈ I ( 1 − a ¯ i ) ⟨ Λ i ( K † ) * x , Γ i x ⟩ ≥ 3 4 − 1 4 λ 2 + λ 2 ‖ x ‖ 2 .

It should be pointed out that the inequalities in Theorems D and E depend entirely on the K -dual pair, there are no corresponding results on this topic depending only on the K - g -frames themselves, and that the inequalities shown in Theorems D and E are one-sided. We also observe that though bilateral inequalities for some other frame versions do emerge recently [17,20–24], they are of the same form. This motivates us, in this work, to explore bilateral inequalities with new forms for K - g -frames, which are induced by either the K - g -frames themselves or other derivatives of K - g -frames. In Section 2, we establish several bilateral inequalities for K - g -frames in closed subspaces by using the idea of transformation, where only the K - g -frames themselves are involved. Some bilateral inequalities for K - g -frames in closed subspaces with the help of the K -dual pair are also given in Section 3. It is worth remarking that the inequalities we presented admit novel structures compared to existing ones concerned with this object and that new types of inequalities for some other generalized frames such as fusion frames, g -frames, continuous fusion frames, Hilbert-Schmidt frames, and continuous g -frames, etc. can be naturally obtained following the approaches of our results.

2 Inequalities induced by the K - g -frames themselves

The following lemma is needed to state the main results of this article.

Lemma 2.1

(See [21, Lemma 2.6]) Suppose that Θ , Φ ∈ ℒℬ ( ℳ ) satisfy Θ + Φ = Id ℳ . Then, for each x ∈ ℳ and each λ ∈ R , we have

‖ Θ x ‖ 2 + 2 λ Re ⟨ Φ x , x ⟩ = ‖ Φ x ‖ 2 + 2 ( 1 − λ ) Re ⟨ Θ x , x ⟩ + ( 2 λ − 1 ) ‖ x ‖ 2 ≥ ( 2 λ − λ 2 ) ‖ x ‖ 2 .

We are now ready to establish some bilateral inequalities for K - g -frames in closed subspaces, by means of the notations S Λ σ , S Λ σ c , and S Λ ′ given above.

Theorem 2.2

Let K ∈ ℒℬ ( ℳ ) and assume that { Λ i } i ∈ I is a K-g-frame for ℳ . If Ran ( K ) is closed, then for every x ∈ Ran ( K ) , for any λ ∈ [ 1 , + ∞ ) and any σ ⊂ I , we have

(2.1) ∑ i ∈ σ ‖ Λ i x ‖ 2 − λ ∑ i ∈ σ c ‖ Λ i x ‖ 2 ≤ ∑ i ∈ I ‖ ( Λ i ∣ Ran ( K ) ) S Λ ′ − 1 S Λ σ x ‖ 2 − λ ∑ i ∈ I ‖ ( Λ i ∣ Ran ( K ) ) S Λ ′ − 1 S Λ σ c x ‖ 2 ≤ ( λ 3 − λ 2 + 1 ) ∑ i ∈ σ ‖ Λ i x ‖ 2 + ( λ 3 − 3 λ 2 + 2 λ − 1 ) ∑ i ∈ σ c ‖ Λ i x ‖ 2 .

Proof

We define a linear bounded operator as follows:

T : Ran ( K ) → ℓ 2 ( ⊕ i ∈ I N i ) , T ( x ) = { Λ i ∣ Ran ( K ) x } i ∈ I .

Then, it is easy to check that the g -frame operator of Λ ′ = { Λ i ∣ Ran ( K ) } i ∈ I is S Λ ′ = T * T . Notice that, as in the classical case, the operator Q = T S Λ ′ T * ∈ ℒ ( ℓ 2 ( ⊕ i ∈ I N i ) ) is the orthogonal projection onto Ran ( K ) .

Now, for any σ ⊂ I , we denote by P σ the orthogonal projection in ℓ 2 ( ⊕ i ∈ I N i ) given by

P σ ( { y i } i ∈ I ) = { z i } i ∈ I where z i = y i if i ∈ σ , z i = 0 if i ∈ σ c .

Since S Λ σ = T * P σ T , the inequalities stated in (2.1) can be rewritten as

(2.2) ‖ P σ T x ‖ 2 − λ ‖ P σ ⊥ T x ‖ 2 ≤ ‖ Q P σ T x ‖ 2 − λ ‖ Q P σ ⊥ T x ‖ 2 ≤ ( λ 3 − λ 2 + 1 ) ‖ P σ T x ‖ 2 + ( λ 3 − 3 λ 2 + 2 λ − 1 ) ‖ P σ ⊥ T x ‖ 2

for x ∈ Ran ( K ) . Denote by y = T x and normalize it to ‖ y ‖ = 1 . Notice that we have Q y = y and the left-hand inequality for λ ≥ 1 follows from the equality

(2.3) ‖ P σ y ‖ 2 − ‖ Q P σ y ‖ 2 = ‖ Q ⊥ P σ y ‖ 2 = ‖ Q ⊥ P σ ⊥ y ‖ 2 = ‖ P σ ⊥ y ‖ 2 − ‖ Q P σ ⊥ y ‖ 2 .

For the inequality on the right, we know, by combining (2.2) with (2.3), that it is equivalent to

(2.4) ( 1 − λ ) ‖ Q P σ ⊥ y ‖ 2 ≤ ( 1 − λ ) [ 2 λ ‖ P σ ⊥ y ‖ 2 − λ 2 ] .

To prove (2.4), it is equivalent to prove that, for λ ≥ 1 , the quadratic function f ( λ ) = λ 2 − 2 λ ‖ P σ ⊥ y ‖ 2 + ‖ Q P σ ⊥ y ‖ 2 ≥ 0 . Since the minimizer for f ( λ ) is in λ 0 = ‖ P σ ⊥ y ‖ 2 and ‖ P σ ⊥ y ‖ 2 = ⟨ Q P σ ⊥ y , y ⟩ ≤ ‖ Q P σ ⊥ y ‖ , we arrive at

f ( λ ) ≥ f ( λ 0 ) = ( ‖ Q P σ ⊥ y ‖ − ‖ P σ ⊥ y ‖ 2 ) ( ‖ Q P σ ⊥ y ‖ + ‖ P σ ⊥ y ‖ 2 ) ≥ 0 .

This completes the proof.□

Theorem 2.3

Let K ∈ ℒℬ ( ℳ ) , and assume that { Λ i } i ∈ I is a K-g-frame for ℳ . If Ran ( K ) is closed, then for each x ∈ Ran ( K ) , for any λ ∈ [ 1 2 , + ∞ ) and any σ ⊂ I , we have

(2.5) ( 4 λ − 1 ) ∑ i ∈ I ‖ ( Λ i ∣ Ran ( K ) ) S Λ ′ − 1 S Λ σ c x ‖ 2 + ( 1 − λ 2 ) ∑ i ∈ I ‖ Λ i x ‖ 2 ≤ ∑ i ∈ σ ‖ Λ i x ‖ 2 + ( 1 + 2 λ ) ∑ i ∈ σ c ‖ Λ i x ‖ 2 ≤ ∑ i ∈ I ‖ ( Λ i ∣ Ran ( K ) ) S Λ ′ − 1 S Λ σ c x ‖ 2 + ( 1 + λ 2 ) ∑ i ∈ I ‖ Λ i x ‖ 2 .

Proof

Drawing support from the notations introduced in the proof of Theorem 2.2, the inequalities in (2.5) can be rewritten as

(2.6) ( 4 λ − 1 ) ‖ Q P σ ⊥ y ‖ 2 + ( 1 − λ 2 ) ≤ ‖ P σ y ‖ 2 + ( 1 + 2 λ ) ‖ P σ ⊥ y ‖ 2 ≤ ‖ Q P σ ⊥ y ‖ 2 + ( 1 + λ 2 )

for λ ≥ 1 2 . It is easy to see that the inequality on the left is equivalent to prove that, for λ ≥ 1 2 ,

g ( λ ) = λ 2 + 2 λ ( ‖ P σ ⊥ y ‖ 2 − 2 ‖ Q P σ ⊥ y ‖ 2 ) + ‖ Q P σ ⊥ y ‖ 2 ≥ 0 ,

which, actually, follows from the fact that g ( 1 2 ) ≥ 0 and g ′ ( 1 2 ) ≥ 0 .

As for the right-hand inequality in (2.6), it is equivalent to show that

0 ≤ f ( λ ) = ( 1 + λ 2 ) + ‖ Q P σ ⊥ y ‖ 2 − ‖ P σ y ‖ 2 − ( 1 + 2 λ ) ‖ P σ ⊥ y ‖ 2 = λ 2 − 2 λ ‖ P σ ⊥ y ‖ 2 + ( 1 − ‖ P σ y ‖ 2 − ‖ P σ ⊥ y ‖ 2 ) + ‖ Q P σ ⊥ y ‖ 2 = λ 2 − 2 λ ‖ P σ ⊥ y ‖ 2 + ‖ Q P σ ⊥ y ‖ 2 ,

which has already been presented in the proof of Theorem 2.2, and we are done.□

Theorem 2.4

Let K ∈ ℒℬ ( ℳ ) , and assume that { Λ i } i ∈ I is a K-g-frame for ℳ . If Ran ( K ) is closed, then for each x ∈ Ran ( K ) , for any λ ∈ R and any σ ⊂ I , we have

(2.7) ( 1 + 2 λ ) ∑ i ∈ σ c ‖ Λ i x ‖ 2 − ( 1 + λ 2 ) ∑ i ∈ I ‖ Λ i x ‖ 2 ≤ ∑ i ∈ I ‖ ( Λ i ∣ Ran ( K ) ) S Λ ′ − 1 S Λ σ c x ‖ 2 − ∑ i ∈ σ ‖ Λ i x ‖ 2 ≤ ( 3 − 2 λ ) ∑ i ∈ σ c ‖ Λ i x ‖ 2 + ( λ 2 − 1 ) ∑ i ∈ I ‖ Λ i x ‖ 2 .

Proof

For any σ ⊂ I , we see from (1.7) that

S Λ ′ − 1 2 S Λ σ S Λ ′ − 1 2 + S Λ ′ − 1 2 S Λ σ c S Λ ′ − 1 2 = S Λ ′ − 1 2 S Λ ′ S Λ ′ − 1 2 = Id Ran ( K ) .

If we let W = S Λ ′ − 1 2 S Λ σ S Λ ′ − 1 2 and T = S Λ ′ − 1 2 S Λ σ c S Λ ′ − 1 2 , then it is easily seen that ⟨ W x , x ⟩ ≥ 0 and ⟨ T x , x ⟩ ≥ 0 for each x ∈ Ran ( K ) , and that W T = T W . Hence,

0 ≤ W T = ( Id Ran ( K ) − T ) T = T − T 2 = S Λ ′ − 1 2 ( S Λ σ c − S Λ σ c S Λ ′ − 1 S Λ σ c ) S Λ ′ − 1 2 ,

meaning that S Λ σ c ≥ S Λ σ c S Λ ′ − 1 S Λ σ c . A similar discussion can show that S Λ σ ≥ S Λ σ S Λ ′ − 1 S Λ σ .

Letting Θ = T and Φ = W , and taking S Λ ′ 1 2 x instead of x in Lemma 2.1 gives

(2.8) ⟨ S Λ ′ − 1 S Λ σ c x , S Λ σ c x ⟩ = ⟨ S Λ ′ − 1 2 S Λ σ c S Λ ′ − 1 2 S Λ ′ 1 2 x , S Λ ′ − 1 2 S Λ σ c S Λ ′ − 1 2 S Λ ′ 1 2 x ⟩ ≥ ( 2 λ − λ 2 ) ⟨ S Λ ′ 1 2 x , S Λ ′ 1 2 x ⟩ − λ ( ⟨ S Λ ′ − 1 2 S Λ σ S Λ ′ − 1 2 S Λ ′ 1 2 x , S Λ ′ 1 2 x ⟩ + ⟨ S Λ ′ 1 2 x , S Λ ′ − 1 2 S Λ σ S Λ ′ − 1 2 S Λ ′ 1 2 x ⟩ ) = ( 2 λ − λ 2 ) ⟨ S Λ ′ x , x ⟩ − 2 λ ⟨ S Λ σ x , x ⟩ = ( 2 λ − λ 2 ) ⟨ S Λ σ c x , x ⟩ − λ 2 ⟨ S Λ σ x , x ⟩

for each x ∈ Ran ( K ) and each λ ∈ R . Therefore,

(2.9) ∑ i ∈ I ‖ ( Λ i ∣ Ran ( K ) ) S Λ ′ − 1 S Λ σ c x ‖ 2 − ∑ i ∈ σ ‖ Λ i x ‖ 2 = ⟨ S Λ ′ − 1 S Λ σ c x , S Λ σ c x ⟩ − ⟨ S Λ σ x , x ⟩ ≥ ( 2 λ − λ 2 ) ⟨ S Λ σ c x , x ⟩ − λ 2 ⟨ S Λ σ x , x ⟩ − ⟨ S Λ σ x , x ⟩ = ( 2 λ − λ 2 ) ⟨ S Λ σ c x , x ⟩ − ( 1 + λ 2 ) ⟨ S Λ ′ x , x ⟩ + ( 1 + λ 2 ) ⟨ S Λ σ c x , x ⟩ = ( 1 + 2 λ ) ⟨ S Λ σ c x , x ⟩ − ( 1 + λ 2 ) ⟨ S Λ ′ x , x ⟩ = ( 1 + 2 λ ) ∑ i ∈ σ c ‖ Λ i x ‖ 2 − ( 1 + λ 2 ) ∑ i ∈ I ‖ Λ i x ‖ 2 .

Again by Lemma 2.1 (replacing Θ , Φ , and x , respectively, by S Λ ′ − 1 2 S Λ σ c S Λ ′ − 1 2 , S Λ ′ − 1 2 S Λ σ S Λ ′ − 1 2 , and S Λ ′ 1 2 x ), it follows that

⟨ S Λ ′ − 1 S Λ σ x , S Λ σ x ⟩ = ⟨ S Λ ′ − 1 2 S Λ σ S Λ ′ − 1 2 S Λ ′ 1 2 x , S Λ ′ − 1 2 S Λ σ S Λ ′ − 1 2 S Λ ′ 1 2 x ⟩ ≥ ( ( 2 λ − λ 2 ) − ( 2 λ − 1 ) ) ⟨ S Λ ′ 1 2 x , S Λ ′ 1 2 x ⟩ − ( 1 − λ ) ( ⟨ S Λ ′ − 1 2 S Λ σ c S Λ ′ − 1 2 S Λ ′ 1 2 x , S Λ ′ 1 2 x ⟩ + ⟨ S Λ ′ 1 2 x , S Λ ′ − 1 2 S Λ σ c S Λ ′ − 1 2 S Λ ′ 1 2 x ⟩ ) = ( 1 − λ 2 ) ⟨ S Λ ′ x , x ⟩ − 2 ( 1 − λ ) ⟨ S Λ σ c x , x ⟩ ,

leading to

⟨ S Λ ′ − 1 S Λ σ c x , S Λ σ c x ⟩ − ⟨ S Λ σ x , x ⟩ ≤ ⟨ S Λ σ c x , x ⟩ − ⟨ S Λ ′ − 1 S Λ σ x , S Λ σ x ⟩ ≤ ⟨ S Λ σ c x , x ⟩ − ( 1 − λ 2 ) ⟨ S Λ ′ x , x ⟩ + 2 ( 1 − λ ) ⟨ S Λ σ c x , x ⟩ = ( 3 − 2 λ ) ⟨ S Λ σ c x , x ⟩ − ( 1 − λ 2 ) ⟨ S Λ x , x ⟩

for any x ∈ Ran ( K ) and any λ ∈ R . That is,

∑ i ∈ I ‖ ( Λ i ∣ Ran ( K ) ) S Λ ′ − 1 S Λ σ c x ‖ 2 − ∑ i ∈ σ ‖ Λ i x ‖ 2 ≤ ( 3 − 2 λ ) ∑ i ∈ σ c ‖ Λ i x ‖ 2 + ( λ 2 − 1 ) ∑ i ∈ I ‖ Λ i x ‖ 2 .

This along with (2.9) gives (2.7).□

3 Inequalities based on a K -dual pair

Assume that K ∈ ℒℬ ( ℳ ) , and that { Λ i } i ∈ I is a K - g -frame for ℳ with a dual K - g -Bessel sequence { Γ i } i ∈ I . Then, there are always two linear bounded operators, associated with the K -dual pair ( { Λ i } i ∈ I , { Γ i } i ∈ I ) and β = { β i } i ∈ I ∈ ℓ ∞ ( I ) , given below

(3.1) U β , U 1 − β : ℳ → ℳ , U β x = ∑ i ∈ I β i Λ i * Γ i x , U 1 − β x = ∑ i ∈ I ( 1 − β i ) Λ i * Γ i x , ∀ x ∈ ℳ .

In what follows, we will state several bilateral inequalities for K - g -frames in closed subspaces by using U β and U 1 − β .

Theorem 3.1

Suppose K ∈ ℒℬ ( ℳ ) . Let { Λ i } i ∈ I be a K-g-frame for ℳ with a dual K-g-Bessel sequence { Γ i } i ∈ I , and let β = { β i } i ∈ I ∈ ℓ ∞ ( I ) be given. If Ran ( K ) is closed, then for each x ∈ Ran ( K ) and each λ ∈ [ 0,2 ] , we have

(3.2) ( λ − λ 2 ) ‖ x ‖ 2 − λ Re ∑ i ∈ I ( 1 − β i ) ⟨ Γ i K † x , Λ i x ⟩ ≤ ∑ i ∈ I β i Λ i * Γ i K † x 2 − λ Re ∑ i ∈ I β i ⟨ Γ i K † x , Λ i x ⟩ ≤ ( 2 − λ ) ‖ U β ‖ 2 ‖ K † ‖ 2 + λ ( ‖ U 1 − β ‖ 2 ‖ K † ‖ 2 − 1 ) 2 ‖ x ‖ 2 .

Proof

It is obvious that U β + U 1 − β = K . Thus,

U β K † + U 1 − β K † = K K † = Pr ( Ran ( K ) ) ,

where Pr ( W ) denotes the orthogonal projection onto the subspace W of ℳ . It follows that

U β K † ∣ Ran ( K ) + U 1 − β K † ∣ Ran ( K ) = Id Ran ( K ) .

Since

− λ Re ⟨ U β K † x , x ⟩ = λ ‖ U β K † x − x ‖ 2 − λ ‖ U β K † x ‖ 2 − λ ‖ x ‖ 2 2

and

‖ U β K † x − x ‖ 2 = ‖ ( U β − K ) K † x ‖ 2 = ‖ − U 1 − β K † x ‖ 2 = ‖ U 1 − β K † x ‖ 2

for any x ∈ Ran ( K ) and any λ ∈ [ 0,2 ] , we have

∑ i ∈ I β i Λ i * Γ i K † x 2 − λ Re ∑ i ∈ I β i ⟨ Γ i K † x , Λ i x ⟩ = ‖ U β K † x ‖ 2 − λ Re ⟨ U β K † x , x ⟩ = ‖ U β K † x ‖ 2 + λ ‖ U 1 − β K † x ‖ 2 − λ ‖ U β K † x ‖ 2 − λ ‖ x ‖ 2 2 = 2 ‖ U β K † x ‖ 2 + λ ‖ U 1 − β K † x ‖ 2 − λ ‖ U β K † x ‖ 2 − λ ‖ x ‖ 2 2 ≤ ( 2 − λ ) ‖ U β ‖ 2 ‖ K † ‖ 2 + λ ( ‖ U 1 − β ‖ 2 ‖ K † ‖ 2 − 1 ) 2 ‖ x ‖ 2 .

To complete the proof, it remains to prove the left-hand inequality of (3.2). By Lemma 2.1, we obtain

‖ U β K † x ‖ 2 ≥ ( 2 λ − λ 2 ) ‖ x ‖ 2 − 2 λ Re ⟨ U 1 − β K † x , x ⟩

for any x ∈ Ran ( K ) . Therefore,

∑ i ∈ I β i Λ i * Γ i K † x 2 − λ Re ∑ i ∈ I β i ⟨ Γ i K † x , Λ i x ⟩ = ‖ U β K † x ‖ 2 − λ Re ⟨ U β K † x , x ⟩ ≥ ( 2 λ − λ 2 ) ‖ x ‖ 2 − 2 λ Re ⟨ U 1 − β K † x , x ⟩ − λ Re ⟨ U β K † x , x ⟩ = ( λ − λ 2 ) ‖ x ‖ 2 + λ 2 ⟨ ( U β K † + U 1 − β K † ) x , x ⟩ + λ 2 ⟨ x , ( U β K † + U 1 − β K † ) x ⟩ − 2 λ Re ⟨ U 1 − β K † x , x ⟩ − λ 2 ⟨ U β K † x , x ⟩ − λ 2 ⟨ x , U β K † x ⟩ = ( λ − λ 2 ) ‖ x ‖ 2 − 2 λ Re ⟨ U 1 − β K † x , x ⟩ + λ 2 ⟨ U 1 − β K † x , x ⟩ + λ 2 ⟨ x , U 1 − β K † x ⟩ = ( λ − λ 2 ) ‖ x ‖ 2 − λ Re ⟨ U 1 − β K † x , x ⟩ = ( λ − λ 2 ) ‖ x ‖ 2 − λ Re ∑ i ∈ I ( 1 − β i ) ⟨ Γ i K † x , Λ i x ⟩ ,

as desired.□

Theorem 3.2

Suppose K ∈ ℒℬ ( ℳ ) . Let { Λ i } i ∈ I be a K-g-frame for ℳ with a dual K-g-Bessel sequence { Γ i } i ∈ I , and let β = { β i } i ∈ I ∈ ℓ ∞ ( I ) be given. If Ran ( K ) is closed, then for every x ∈ Ran ( K ) and every λ ∈ [ 0,1 ] , we have

(3.3) ( 2 λ − λ 2 ) ‖ x ‖ 2 ≤ ∑ i ∈ I β i Λ i * Γ i K † x 2 + 2 λ Re ∑ i ∈ I ( 1 − β i ) ⟨ Γ i K † x , Λ i x ⟩ ≤ ( ( ( 1 − λ ) ‖ U β ‖ 2 + λ ‖ U 1 − β ‖ 2 ) ‖ K † ‖ 2 + λ ) ‖ x ‖ 2 .

Proof

The left-hand inequality follows directly from Lemma 2.1, if we take U β K † and U 1 − β K † instead of Θ and Φ , respectively. For the inequality on the right-hand side, we have for each x ∈ Ran ( K ) and each λ ∈ [ 0,1 ] that

∑ i ∈ I β i Λ i * Γ i K † x 2 + 2 λ Re ∑ i ∈ I ( 1 − β i ) ⟨ Γ i K † x , Λ i x ⟩ = ‖ U β K † x ‖ 2 + 2 λ Re ⟨ U 1 − β K † x , x ⟩ = ‖ U β K † x ‖ 2 + 2 λ ‖ x ‖ 2 − 2 λ Re ⟨ U β K † x , x ⟩

□ = ‖ U β K † x ‖ 2 + 2 λ ‖ x ‖ 2 + λ ( ‖ ( U β − K ) K † x ‖ 2 − ‖ U β K † x ‖ 2 − ‖ x ‖ 2 ) = ‖ U β K † x ‖ 2 + 2 λ ‖ x ‖ 2 + λ ( ‖ − U 1 − β K † x ‖ 2 − ‖ U β K † x ‖ 2 − ‖ x ‖ 2 ) = ( 1 − λ ) ‖ U β K † x ‖ 2 + λ ‖ x ‖ 2 + λ ‖ U 1 − β K † x ‖ 2 ≤ ( ( ( 1 − λ ) ‖ U β ‖ 2 + λ ‖ U 1 − β ‖ 2 ) ‖ K † ‖ 2 + λ ) ‖ x ‖ 2 .

Remark 3.3

Actually, the inequality on the left-hand side of (3.3) remains true for any λ ∈ R , which is just Theorem D (1), and we give here, however, an upper bound condition.

Theorem 3.4

Suppose K ∈ ℒℬ ( ℳ ) . Let { Λ i } i ∈ I be a K-g-frame for ℳ with a dual K-g-Bessel sequence { Γ i } i ∈ I , and let β = { β i } i ∈ I ∈ ℓ ∞ ( I ) be given. If Ran ( K ) is closed, then for any x ∈ Ran ( K ) and any λ ≥ 0 , we have

4 λ 2 Re ∑ i ∈ I β i ⟨ Γ i K † x , Λ i x ⟩ − λ 2 ( 1 + 2 λ ) ‖ x ‖ 2 ≤ ∑ i ∈ I β i Λ i * Γ i K † x 2 − 2 λ Re ∑ i ∈ I ( 1 − β i ) ⟨ Γ i K † x , Λ i x ⟩ + 2 λ ∑ i ∈ I ( 1 − β i ) Λ i * Γ i K † x 2 ≤ ( ( ( 1 + λ ) ‖ U β ‖ 2 + λ ‖ U 1 − β ‖ 2 ) ‖ K † ‖ 2 − 1 ) ‖ x ‖ 2 .

Proof

The proof of the right-hand inequality is similar to Theorem 3.1. For the opposite inequality, we have, by Lemma 2.1, that

‖ U 1 − β x ‖ 2 − 2 Re ⟨ U 1 − β x , x ⟩ ≥ ( 1 − λ 2 ) ‖ x ‖ 2 − ( 2 − 2 λ ) Re ⟨ U β x , x ⟩ − 2 Re ⟨ U 1 − β x , x ⟩ = ( 1 − λ 2 ) ‖ x ‖ 2 + 2 λ Re ⟨ U β x , x ⟩ − 2 ( Re ⟨ U β x , x ⟩ + Re ⟨ U 1 − β x , x ⟩ ) = ( 1 − λ 2 ) ‖ x ‖ 2 − 2 ‖ x ‖ 2 + 2 λ Re ⟨ U β x , x ⟩ = 2 λ Re ⟨ U β x , x ⟩ − ( 1 + λ 2 ) ‖ x ‖ 2

for each x ∈ Ran ( K ) and each λ ≥ 0 . Again by Lemma 2.1,

∑ i ∈ I β i Λ i * Γ i K † x 2 − 2 λ Re ∑ i ∈ I ( 1 − β i ) ⟨ Γ i K † x , Λ i x ⟩ + 2 λ ∑ i ∈ I ( 1 − β i ) Λ i * Γ i K † x 2 = ‖ U β x ‖ 2 − 2 λ Re ⟨ U 1 − β x , x ⟩ + 2 λ ‖ U 1 − β x ‖ 2 ≥ ( 2 λ − λ 2 ) ‖ x ‖ 2 − 2 λ Re ⟨ U 1 − β x , x ⟩ − 2 λ Re ⟨ U 1 − β x , x ⟩ + 2 λ ‖ U 1 − β x ‖ 2 = ( 2 λ − λ 2 ) ‖ x ‖ 2 + 2 λ ( ‖ U 1 − β x ‖ 2 − 2 Re ⟨ U 1 − β x , x ⟩ ) ≥ ( 2 λ − λ 2 ) ‖ x ‖ 2 + 2 λ ( 2 λ Re ⟨ U β x , x ⟩ − ( 1 + λ 2 ) ‖ x ‖ 2 ) = 4 λ 2 Re ⟨ U β x , x ⟩ − λ 2 ( 1 + 2 λ ) ‖ x ‖ 2 = 4 λ 2 Re ∑ i ∈ I β i ⟨ Γ i K † x , Λ i x ⟩ − λ 2 ( 1 + 2 λ ) ‖ x ‖ 2 ,

and we arrive at the conclusion.□

Assume that { Λ i } i ∈ I is a K - g -frame for ℳ with a dual K - g -Bessel sequence { Γ i } i ∈ I . For any σ ⊂ I , letting U σ x = ∑ i ∈ σ Λ i * Γ i x and U σ c x = ∑ i ∈ σ c Λ i * Γ i x for each x ∈ ℳ . We can now immediately draw the following conclusion, if in Theorems 3.1, 3.2, and 3.4, we take

β i = 1 if i ∈ σ , 0 if i ∈ σ c .

Corollary 3.5

Suppose K ∈ ℒℬ ( ℳ ) . Let { Λ i } i ∈ I be a K-g-frame for ℳ with a dual K-g-Bessel sequence { Γ i } i ∈ I . If Ran ( K ) is closed, then

(1) For each x ∈ Ran ( K ) , for any λ ∈ [ 0,2 ] and any σ ⊂ I ,

( λ − λ 2 ) ‖ x ‖ 2 − λ Re ∑ i ∈ σ c ⟨ Γ i K † x , Λ i x ⟩ ≤ ∑ i ∈ σ Λ i * Γ i K † x 2 − λ Re ∑ i ∈ σ ⟨ Γ i K † x , Λ i x ⟩ ≤ ( 2 − λ ) ‖ U σ ‖ 2 ‖ K † ‖ 2 + λ ( ‖ U σ c ‖ 2 ‖ K † ‖ 2 − 1 ) 2 ‖ x ‖ 2 .

(2) For every x ∈ Ran ( K ) , for any λ ∈ [ 0,1 ] and any σ ⊂ I ,

( 2 λ − λ 2 ) ‖ x ‖ 2 ≤ ∑ i ∈ σ Λ i * Γ i K † x 2 + 2 λ Re ∑ i ∈ σ c ⟨ Γ i K † x , Λ i x ⟩ ≤ ( ( ( 1 − λ ) ‖ U σ ‖ 2 + λ ‖ U σ c ‖ 2 ) ‖ K † ‖ 2 + λ ) ‖ x ‖ 2 .

(3) For each x ∈ Ran ( K ) , for any λ ≥ 0 and any σ ⊂ I ,

4 λ 2 Re ∑ i ∈ σ ⟨ Γ i K † x , Λ i x ⟩ − λ 2 ( 1 + 2 λ ) ‖ x ‖ 2 ≤ ∑ i ∈ σ Λ i * Γ i K † x 2 − 2 λ Re ∑ i ∈ σ c ⟨ Γ i K † x , Λ i x ⟩ + 2 λ ∑ i ∈ σ c Λ i * Γ i K † x 2 ≤ ( ( ( 1 + λ ) ‖ U σ ‖ 2 + λ ‖ U σ c ‖ 2 ) ‖ K † ‖ 2 − 1 ) ‖ x ‖ 2 .

Suppose that K ∈ ℒℬ ( ℳ ) has closed range, and that { Λ i } i ∈ I is a Parseval K - g -frame for ℳ . From [13, Theorem 2.7], we see that { Λ i ( K † ) * } i ∈ I is a dual K - g -Bessel sequence of { Λ i } i ∈ I . Thus, Theorems 3.1, 3.2, and 3.4 bring a direct consequence as follows.

Corollary 3.6

Suppose that K ∈ ℒℬ ( ℳ ) is a closed range operator, that { Λ i } i ∈ I is a Parseval K-g-frame for ℳ , and that β = { β i } i ∈ I ∈ ℓ ∞ ( I ) . Then, we obtain

(1) For each x ∈ Ran ( K ) and each λ ∈ [ 0,2 ] ,

( λ − λ 2 ) ‖ x ‖ 2 − λ Re ∑ i ∈ I ( 1 − β i ) ⟨ K † x , K † Λ i * Λ i x ⟩ ≤ ∑ i ∈ I β i Λ i * Λ i ( K † ) * K † x 2 − λ Re ∑ i ∈ I β i ⟨ K † x , K † Λ i * Λ i x ⟩ ≤ ( 2 − λ ) ‖ U β ‖ 2 ‖ K † ‖ 2 + λ ( ‖ U 1 − β ‖ 2 ‖ K † ‖ 2 − 1 ) 2 ‖ x ‖ 2 .

(2) For every x ∈ Ran ( K ) and every λ ∈ [ 0,1 ] ,

( 2 λ − λ 2 ) ‖ x ‖ 2 ≤ ∑ i ∈ I β i Λ i * Λ i ( K † ) * K † x 2 + 2 λ Re ∑ i ∈ I ( 1 − β i ) ⟨ K † x , K † Λ i * Λ i x ⟩ ≤ ( ( ( 1 − λ ) ‖ U β ‖ 2 + λ ‖ U 1 − β ‖ 2 ) ‖ K † ‖ 2 + λ ) ‖ x ‖ 2 .

(3) For any x ∈ Ran ( K ) and any λ ≥ 0 ,

4 λ 2 Re ∑ i ∈ I β i ⟨ K † x , K † Λ i * Λ i x ⟩ − λ 2 ( 1 + 2 λ ) ‖ x ‖ 2 ≤ ∑ i ∈ I β i Λ i * Λ i ( K † ) * K † x 2 − 2 λ Re ∑ i ∈ I ( 1 − β i ) ⟨ K † x , K † Λ i * Λ i x ⟩

+ 2 λ ∑ i ∈ I ( 1 − β i ) Λ i * Λ i ( K † ) * K † x 2 ≤ ( ( ( 1 + λ ) ‖ U β ‖ 2 + λ ‖ U 1 − β ‖ 2 ) ‖ K † ‖ 2 − 1 ) ‖ x ‖ 2 .

4 Conclusions

In this work, we establish several bilateral inequalities for K - g -frames in subspaces based on the theory of operators, which admit novel structures compared to previous ones. Particularly, if we take K = Id ℳ , then Ran ( K ) = ℳ and K † = Id ℳ . Therefore, following the approaches of our results, we can easily set up corresponding bilateral inequalities for some other generalized frames such as fusion frames, g -frames, continuous fusion frames, Hilbert-Schmidt frames and continuous g -frames, which possess new types of structures compared with the ones given in the articles [17–24].

Acknowledgements

The author thanks the reviewers for their constructive comments which have led to a significant improvement of this work.

Funding information: This work was supported by the National Natural Science Foundation of China (Grant Nos. 12361028 and 11761057).
Author contributions: The author has accepted responsibility for the entire content of this manuscript and approved its submission.
Conflict of interest: The author states no conflict of interest.
Ethical approval: The conducted research is not related to either human or animal use.
Data availability statement: Data sharing is not applicable to this article as no data sets were generated or analyzed during this study.

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Received: 2024-05-22

Revised: 2024-10-21

Accepted: 2024-11-20

Published Online: 2025-03-19

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Keywords for this article

K-dual pair; pseudo-inverse operator; bilateral inequality

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