Unitarily equivalent bilateral weighted shifts with operator weights

Abstract

The aim of this paper is to study unitarily equivalent bilateral weighted shifts with operator weights. Our purpose is to establish a general characterization of unitary equivalence of such shifts under the assumption that the weights are quasi-invertible. Under further assumptions on weights it was proved that unitary equivalence of bilateral weigthed shifts with operator weights defined on $\mathbb{C}^{2}$ can always be given by a unitary operator with at most two non-zero diagonals. The paper contains also examples of unitarily equivalent shifts with weights defined on $\mathbb{C}^{k}$ such that every unitary operator, which intertwines them has at least $k$ non-zero diagonals