Weaving K-Frames in Hilbert Spaces

Abstract

Gǎvruta 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 $A \le B$ 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.