Unitary equivalence of balanced weighted shifts on rooted directed trees

TL;DR

This paper provides a complete characterization of unitary equivalence for non-periodic balanced weighted shifts on rooted directed trees, extending previous results and exploring conditions for periodic cases.

Contribution

It generalizes unitary equivalence criteria for weighted shifts on rooted trees and introduces new conditions under which these shifts are equivalent.

Findings

Characterization of non-periodic balanced weighted shifts under invertibility conditions

Extension of unitary equivalence results to Bergman and Dirichlet type shifts

Counterexample showing criteria are not necessary for eventually periodic shifts

Abstract

We completely characterize non-periodic balanced weighted shifts $S_{\lambdab}$ on rooted directed trees under a very mild assumption that $S_{\lambdab}^{*n}S_{\lambdab}^n|_{\ker S_{\lambdab}^*}$ is invertible operator on $\ker S_{\lambdab}^*$ for all $n \in \mathbb N$. This generalizes the previously established unitary equivalences for Bergman and Dirichlet type shifts associated with locally finite rooted directed trees. We also give a counter example to justify that the criteria obtained for non-periodic balanced weighted shifts is not necessary for eventually periodic balanced weighted shifts.