Weaving K-frames in Hilbert Spaces

TL;DR

This paper investigates the properties of weaving K-frames in Hilbert spaces, establishing conditions for their equivalence and stability under perturbations, thus extending the understanding of atomic systems with respect to bounded operators.

Contribution

It provides necessary and sufficient conditions for weaving K-frames, proves the equivalence of woven and weakly woven K-frames, and offers criteria for their stability under perturbations.

Findings

Woven K-frames and weakly woven K-frames are equivalent.

Necessary and sufficient conditions for weaving K-frames are established.

Perturbation stability of weaving K-frames is characterized.

Abstract

Gavruta introduced $K$-frames for Hilbert spaces to study atomic systems with respect to a bounded linear operator. There are many differences between K-frames and standard frames, so we study weaving properties of K-frames. Two frames $\{\phi_{i}\}_{i \in I}$ and $\{\psi_{i}\}_{i \in I}$ for a separable Hilbert space $\mathcal{H}$ are woven if there are positive constants $A \leq B$ such that for every subset $\sigma \subset I$, the family $\{\phi_{i}\}_{i \in \sigma} \cup \{\psi_{i}\}_{i \in \sigma^{c}}$ is a frame for $\mathcal{H}$ with frame bounds $A, B$. In this paper, we present necessary and sufficient conditions for weaving $K$-frames in Hilbert spaces. It is shown that woven $K$-frames and weakly woven $K$-frames are equivalent. Finally, sufficient conditions for Paley-Wiener type perturbation of weaving $K$-frames are given.