The Wold-type decomposition for $m$-isometries

TL;DR

This paper investigates the Wold-type decomposition for $m$-isometries, providing conditions for analytic cases, characterizations of weighted shifts, and distinctions from composition operators on directed graphs.

Contribution

It introduces the $k$-kernel condition for $m$-isometries, characterizes when they are unitarily equivalent to weighted shifts, and distinguishes certain composition operators from weighted shifts.

Findings

Established an equivalent condition for analytic $m$-isometries to admit Wold-type decomposition.

Characterized $m$-isometric unilateral operator valued weighted shifts with positive, commuting weights.

Showed that certain $m$-isometric composition operators are not unitarily equivalent to unilateral weighted shifts.

Abstract

The aim of this paper is to study the Wold-type decomposition in the class of $m$-isometries. One of our main results establishes an equivalent condition for an analytic $m$-isometry to admit the Wold-type decomposition for $m\ge2$. In particular, we introduce the $k$-kernel condition which we use to characterize analytic $m$-isometric operators which are unitarily equivalent to unilateral operator valued weighted shifts for $m\ge2$. As a result, we also show that $m$-isometric composition operators on directed graphs with one circuit containing only one element are not unitarily equivalent to unilateral weighted shifts. We also provide a characterization of $m$-isometric unilateral operator valued weighted shifts with positive and commuting weights.