Factorable Weak Operator-Valued Frames

Abstract

Let $\mathcal{H}$ and $\mathcal{H}_0$ be Hilbert spaces and $\{A_n\}_n$ be a sequence of bounded linear operators from $\mathcal{H}$ to $\mathcal{H}_0$. The study frames for Hilbert spaces initiated the study of operators of the form $\sum_{n=1}^{\infty}A_n^*A_n$, where the convergence is in the strong-operator topology, by Kaftal, Larson and Zhang in the paper: Operator-valued frames. \textit{Trans. Amer. Math. Soc.}, 361(12):6349-6385, 2009. In this paper, we generalize this and study the series of the form $\sum_{n=1}^{\infty}\Psi_n^*A_n$, where $\{\Psi _n\}_n$ is a sequence of operators from $\mathcal{H}$ to $\mathcal{H}_0$. Main tool used in the study of $\sum_{n=1}^{\infty}A_n^*A_n$ is the factorization of this series. Since the series $\sum_{n=1}^{\infty}\Psi_n^*A_n$ may not be factored, it demands greater care. Therefore we impose a factorization of $\sum_{n=1}^{\infty}\Psi_n^*A_n$ and derive various results. We characterize them and derive dilation results. We further study the series by taking the indexed set as group as well as group-like unitary system. We also derive stability results.