# Unitary equivalence of operator-valued multishifts

## TL;DR

This paper explores the properties and function theory of operator-valued multishifts, including their unitary equivalence, bounded point evaluations, and spectral characteristics, expanding understanding of their structure and behavior.

## Contribution

It introduces new results on the unitary equivalence and spectral properties of operator-valued multishifts, especially with invertible weights, and compares these with classical cases.

## Key findings

- Operator-valued multishifts include multishifts on rooted directed trees.
- Established circularity, analyticity, and wandering subspace properties.
- Bounded point evaluations can be properly contained in the joint point spectrum.

## Abstract

We systematically study various aspects of operator-valued multishifts. Beginning with basic properties, we show that the class of multishifts on the directed Cartesian product of rooted directed trees is contained in that of operator-valued multishifts. Further, we establish circularity, analyticity and wandering subspace property of these multishifts. In the rest part of the paper, we study the function theoretic behaviour of operator-valued multishifts. We determine the bounded point evaluation, reproducing kernel structure and the unitary equivalence of operator-valued multishifts with invertible operator weights. In contrast with a result of Lubin, it appears that the set of all bounded point evaluations of an operator-valued multishift may be properly contained in the joint point spectrum of the adjoint of underlying multishift.