Wold-type decomposition for left-invertible weighted shifts on a rootless directed tree

TL;DR

This paper characterizes when bounded left-invertible weighted shifts on rootless directed trees admit a Wold-type decomposition, linking it to the convergence of a series involving the moments of the shift's adjoint.

Contribution

It provides a necessary and sufficient condition for the existence of Wold-type decomposition for such shifts based on series convergence.

Findings

Characterization of Wold-type decomposition for weighted shifts on rootless trees.

Connection between decomposition and convergence of a specific series involving moments.

Main theorem giving a complete criterion for the decomposition.

Abstract

Let $S_{\lambdab}$ be a bounded left-invertible weighted shift on a rootless directed tree $\mathcal T=(V, \mathcal E).$ We address the question of when $S_{\lambdab}$ has Wold-type decomposition. We relate this problem to the convergence of the series $\displaystyle {\tiny \sum_{n = 1}^{\infty} \sum_{u \in G_{v, n}\backslash G_{v, n-1}} \Big(\frac{\lambdab^{(n)}(u)}{\lambdab^{(n)}(v)}\Big)^2},$ $v \in V,$ involving the moments $\lambdab^{(n)}$ of $S^*_{\lambdab}$, where $G_{v, n}=\childn{n}{\parentn{n}{v}}.$ The main result of this paper characterizes all bounded left-invertible weighted shifts $S_{\lambdab}$ on $\mathcal T,$ which have Wold-type decomposition.