New Inequalities of Weaving K-Frames in Subspaces

In the present paper, we obtain some new inequalities for weaving K-frames in subspaces based on the operator methods. The inequalities are associated with a sequence of bounded complex numbers and a parameter λ ∈ R. We also give a double inequality for weaving K-frames with the help of two bounded linear operators induced by K-dual. Facts prove that our results cover those recently obtained on weaving frames due to Li and Leng, and Xiang.


Introduction
This paper adopts the following notations: J is a countable index set, H and K are complex Hilbert spaces, and Id H and R are used to denote respectively the identical operator on H and the set of real numbers. As usual, we denote by B(H, K) the set of all bounded linear operators on H and, if H = K, then B(H, K) is abbreviated to B(H).
Frames were introduced by Duffin and Schaeffer [1] in their study of nonharmonic Fourier series, which have now been used widely not only in theoretical work [2,3], but also in many application areas such as quantum mechanics [4], sampling theory [5][6][7], acoustics [8], and signal processing [9]. As a generalization of frames, the notion of K-frames (also known as frames for operators) was proposed by L. Gȃvruţa [10] when dealing with atomic decompositions for a bounded linear operator K. Please check the papers [11][12][13][14][15][16][17] for further information of K-frames.
Recall that a family {ψ j } j∈J ⊂ H is called a K-frame for H, if there exist two positive numbers A and B satisfying The constants A and B are called K-frame bounds. If K = Id H , then a K-frame turns to be a frame. In addition, if only the right-hand inequality holds, then we call {ψ j } j∈J a Bessel sequence.
Inspired by a question arising in distributed signal processing, Bemrose et al. [18] introduced the concept of weaving frames, which have interested many scholars because of their potential applications such as in wireless sensor networks and pre-processing of signals; see [19][20][21][22][23][24]. Later on, Deepshikha and Vashisht [25] applied the idea of L. Gȃvruţa to the case of weaving frames and thus providing us the notion of weaving K-frames.
Balan et al. [26] obtained an interesting inequality when they further examined the remarkable identity for Parseval frames deriving from their work on signal reconstruction [27]. The inequality was then extended to alternate dual frames and general frames by P. Gȃvruţa [28], the results in which have already been applied in quantum information theory [29]. Recently, those inequalities have been extended to some generalized versions of frames such as continuous g-frames [30], fusion frames and continuous fusion frames [31,32], Hilbert-Schmidt frames [33], and weaving frames [34,35].
Let Ψ 1 = {ψ 1j } j∈J be a given K-frame for H. For any σ ⊂ J, we can define a positive operator S σ Ψ 1 in the following way: In the following, we show that, for given two K-woven frames, we can get some inequalities under the condition that K has a closed range, which are related to a sequence of bounded complex numbers, the corresponding K-dual and a parameter λ ∈ R. Theorem 1. Suppose that K ∈ B(H) has a closed range and K-frames Ψ 1 = {ψ 1j } j∈J and Ψ 2 = {ψ 2j } j∈J in H are K-woven. Then, (i) for any f ∈ Range(K), for all σ ⊂ J, {a j } j∈J ∈ ∞ (J), and λ ∈ R, Proof. We define two bounded linear operators P 1 and P 2 on H as follows: Then, clearly, P 1 f + P 2 f = K f for each f ∈ H and thus P 1 + P 2 = K. Since K has a closed range, by Lemma 2, we have where P Range(K) is the orthogonal projection onto Range(K). Thus, By Lemma 3 (taking λ 2 instead of λ), we get for any f ∈ Range(K). Hence, It follows that from which we arrive at For the inequality in Equation (1), we apply Lemma 3 again, Thus, for any f ∈ Range(K), The proof is similar to (i), so we omit the details.
Proof. Letting K † = Id H and In addition, taking S −1/2 A direction calculation shows that and, similarly, Thus, the result follows if, in Equation (5), we take a j = 1, j ∈ σ, 0, j ∈ σ c .
Proof. For any f ∈ Range(K), for all σ ⊂ J, {a j } j∈J ∈ ∞ (J), and λ ∈ R, we know, by combining Equation (3) and Lemma 3, that For the "Moreover" part, we have for any f ∈ Range(K) that With a similar discussion, we can show that Corollary 4. Suppose that two frames Ψ 1 = {ψ 1j } j∈J and Ψ 2 = {ψ 2j } j∈J in H are woven. Then, for any σ ⊂ J, for all λ ∈ R and all f ∈ H, we have Proof. Letting K † = Id H and for any σ ⊂ J, taking If, now, we replace ψ 1j , ψ 2j and f in the left-hand inequality of Theorem 2 respectively by S −1/2 This along with Equations (6) and (7) gives the left-hand inequality in Equation (8), and the proof of the right-hand inequality is similar and we omit the details. Theorem 3. Suppose that K ∈ B(H) has a closed range and that K-frames Ψ 1 = {ψ 1j } j∈J and Ψ 2 = {ψ 2j } j∈J in H are K-woven. Then, for all σ ⊂ J, for any {a j } j∈J ∈ ∞ (J), λ ∈ R and f ∈ Range(K), for any f ∈ Range(K), where P 1 and P 2 are given in Equation (2).
Proof. For all σ ⊂ J, for any {a j } j∈J ∈ ∞ (J), λ ∈ R and f ∈ Range(K), we see from Equation (4) that Corollary 5. Let the two frames Ψ 1 = {ψ 1j } j∈J and Ψ 2 = {ψ 2j } j∈J in H be woven. Then, for any σ ⊂ J, for all λ ∈ R and all f ∈ H, we have Proof. The proof is similar to Corollary 4 by using Theorem 3, so we omit it.
We conclude the paper with a double inequality for K-weaving frames stated as follows.
Proof. For any σ ⊂ J, for all {a j } j∈J ∈ ∞ (J) and all f ∈ H, it is easy to check that P 1 + P 2 = K. By Lemma 1, we get We also have and the proof is over.

Conflicts of Interest:
The author declares no conflict of interest.