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 ${\sum }_{n=1}^{\infty }{A}_n^{\ast }{A}_n$, 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 ${\sum }_{n=1}^{\infty }{\Psi }_n^{\ast }{A}_n$, where $\{{\Psi }_n{\}}_n$ is a sequence of operators from $\mathcal{H}$ to ${\mathcal{H}}_0$. Main tool used in the study of ${\sum }_{n=1}^{\infty }{A}_n^{\ast }{A}_n$ is the factorization of this series. Since the series ${\sum }_{n=1}^{\infty }{\Psi }_n^{\ast }{A}_n$ may not be factored, it demands greater care. Therefore we impose a factorization of ${\sum }_{n=1}^{\infty }{\Psi }_n^{\ast }{A}_n$ 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.

This paper contains 24 pages and it is an improved version of a part of the unpublished paper arXiv:1810.01629v1 of the same authors