Weaving K-frames in Hilbert Spaces

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 $\{{\varphi }_i{\}}_{i\in I}$ and $\{{\psi }_i{\}}_{i\in I}$ for a separable Hilbert space $\mathcal{H}$ are woven if there are positive constants $A \leq B$suchthatforeverysubset$\sigma \subset I$,thefamily${\phi_{i}}{i \in \sigma} \cup {\psi{i}}_{i \in \sigma^{c}}$isaframefor$$\mathcal{H}$$withframebounds$A, B$. 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.