m-isometric weighted shifts with operator weights

TL;DR

This paper characterizes m-isometric weighted shifts with operator weights, providing construction methods, conditions for commuting weights, and solutions to the completion problem, with applications to bilateral shifts and illustrative examples.

Contribution

It introduces a polynomial operator coefficient characterization and construction procedures for m-isometric weighted shifts, advancing understanding of their structure and properties.

Findings

Characterization of m-isometric shifts via polynomials with operator coefficients

Construction method for all m-isometric shifts with positive weights

Conditions for commuting weights and solutions to the completion problem

Abstract

The aim of this paper is to study $ m $-isometric weighted shifts with operator weights (both unilateral and bilateral). We obtain a characterization of such shifts by polynomials with operator coefficients. The procedure of construction of all $ m $-isometric weighted shifts with positive weights is given. We answer the question when the weights of $ m $-isometric shifts are commuting. The completion problem for $ m $-isometric weighted shifts with operator weights is solved. We characterize $ m $-isometric bilateral shifts the adjoints of which are also $ m $-isometric. Several relevant examples are provided.