Complete systems of unitary invariants for some classes of $2$-isometries

TL;DR

This paper characterizes the unitary equivalence of certain 2-isometric operators satisfying the kernel condition, providing a graph-theoretic complete invariant system for weighted shifts on rooted directed trees.

Contribution

It introduces a model based on operator valued unilateral weighted shifts and offers a graph-theoretic invariant system for classifying 2-isometric weighted shifts on rooted trees.

Findings

Complete system of invariants in graph-theoretic terms.

Characterization of unitary equivalence for 2-isometries with kernel condition.

Analysis of Cauchy dual operators in specific classes.

Abstract

The unitary equivalence of $2$-isometric operators satisfying the so-called kernel condition is characterized. It relies on a model for such operators built on operator valued unilateral weighted shifts and on a characterization of the unitary equivalence of operator valued unilateral weighted shifts in a fairly general context. A complete system of unitary invariants for $2$-isometric weighted shifts on rooted directed trees satisfying the kernel condition is provided. It is formulated purely in the langauge of graph-theory, namely in terms of certain generation branching degrees. The membership of the Cauchy dual operators of $2$-isometries in classes $C_{0 \cdot}$ and $C_{\cdot 0}$ is also studied.