On unitary equivalence of bilateral operator valued weighted shifts

TL;DR

This paper characterizes when two bilateral operator valued weighted shifts are unitarily equivalent using diagonal operators, providing examples and exploring the nature of such unitary operators.

Contribution

It introduces a new characterization of unitary equivalence for bilateral operator valued weighted shifts with quasi-invertible weights and examines the structure of the unitary operators involved.

Findings

Unitary equivalence characterized by diagonal operators for certain shifts

Existence of shifts unitarily equivalent without diagonal operators

Analysis of the structure of unitary operators inducing equivalence

Abstract

We establish a characterization of unitary equivalence of two bilateral operator valued weighted shifts with quasi-invertible weights by an operator of diagonal form. We also present an example of unitary equivalence between shifts defined on $\mathbb{C}^2$ which cannot be given by any unitary operator of diagonal form. The paper is concluded with investigation of unitary operators than can give unitary equivalence of bilateral operator valued weighted shifts.