Factorable Weak Operator-Valued Frames

DOI10.1007/S43034-021-00155-4arXiv2011.05875MaRDI QIDQ6353532

Publication date: 11 November 2020

Abstract: Let

m a t h c a l H

and

m a t h c a l H_{0}

be Hilbert spaces and

{A_{n}}_{n}

be a sequence of bounded linear operators from

m a t h c a l H

to

m a t h c a l H_{0}

. The study frames for Hilbert spaces initiated the study of operators of the form

s u m_{n = 1}^{i n f t y} A_{n}^{*} A_{n}

, where the convergence is in the strong-operator topology, by Kaftal, Larson and Zhang in the paper: Operator-valued frames. extit{Trans. Amer. Math. Soc.}, 361(12):6349-6385, 2009. In this paper, we generalize this and study the series of the form

s u m_{n = 1}^{i n f t y} P s i_{n}^{*} A_{n}

, where

{P s i_{n}}_{n}

is a sequence of operators from

m a t h c a l H

to

m a t h c a l H_{0}

. Main tool used in the study of

s u m_{n = 1}^{i n f t y} A_{n}^{*} A_{n}

is the factorization of this series. Since the series

s u m_{n = 1}^{i n f t y} P s i_{n}^{*} A_{n}

may not be factored, it demands greater care. Therefore we impose a factorization of

s u m_{n = 1}^{i n f t y} P s i_{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.

Mathematics Subject Classification ID

Several-variable operator theory (spectral, Fredholm, etc.) (47A13) Perturbation theory of linear operators (47A55) General harmonic expansions, frames (42C15)

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