Frame sequences and dual frames for operators

Abstract

ABSTRACT:We introduce the more general frame sequences and dual frames related to a linear bounded operator K in Hilbert spaces which we call K-frame sequences and dual K-frames, respectively. We give several equivalent characterizations for K-frame sequences. We also investigate the relationships among K-frame sequences, K-frames, and frame sequences, and give a new perturbation result for K-frames by using the associated dual K-frames. It turns out that in many ways K-frame sequences and dual K-frames behave completely differently from frame sequences and dual frames, respectively.

Weaving of K-g-frames in Hilbert spaces

“…where the first inequality is deduced by (7) and that {Λ j : j ∈ J} and {Γ j : j ∈ J} are K-woven with the universal bounds A and B, and the third inequality is deduced by (6). On the other hand, we have…”

“…Here, A and B are the lower and upper frame bounds, respectively, for the g-frame {Λ j : j ∈ J}. Xiao et al 2 further generalized g-frames [3][4][5] and Kframes [6][7][8] and introduced another notion, namely, K-g-frames. For more information on g-frames and K-g-frames, see Refs.…”

New Inequalities of Weaving K-Frames in Subspaces

“…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.…”

$\boldsymbol{K}$-frames and $\boldsymbol{K}$-Riesz bases incomplex Hilbert spaces

“…架和某些特殊的 K-框架 (如 K-Riesz 框架和 K-Riesz 基等) 的一些性质是非常有意义的. 许多国内 外专家学者已经得到了 Hilbert 空间中的 K-框架、K-Riesz 框架或 K-Riesz 基的一些重要的性质 (参 见文献 [7,[9][10][11][12][13][14][15] 最后, 考虑 Hilbert 空间 H 中的 K-框架或 K-Riesz 基的扰动的稳定性. 文献 [15]…”