PERMUTATIONS AND WOVEN g-FRAMES

Abstract

Two $g$-frames ${\{{\mathrm{\Lambda }}_i\}}_{i\in \mathcal{ℐ}}$ and ${\{{\mathrm{\Gamma }}_i\}}_{i\in \mathcal{ℐ}}$ for the Hilbert space $\mathcal{𝒰}$ are called woven if for each subset $\sigma$ of $\mathcal{ℐ}$ the weaving ${\{{\mathrm{\Lambda }}_i\}}_{i\in \sigma }\cup {\{{\mathrm{\Gamma }}_i\}}_{i\in {\sigma }^c}$ is a $g$-frame for $\mathcal{𝒰}$. The aim of this paper is to consider the reordered families of a $g$-frame ${\{{\mathrm{\Lambda }}_i\}}_{i\in \mathcal{ℐ}}$ with various woven problems. First, we state some useful results for exact $g$-frames and excess of $g$-frames. Then for $\sigma \subset \mathcal{ℐ}$ we consider the families of weavings ${\{{\mathrm{\Lambda }}_i\}}_{i\in \sigma }\cup {\{{\mathrm{\Lambda }}_{\pi (i)}\}}_{i\in {\sigma }^c}$ where $\pi$ is a permutation function on $\mathcal{ℐ}$ and obtain some new conclusions. Lastly, we give relations of reordered weavings and operators, especially $g$-frame operators.