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

TL;DR

Problem: characterize when a bounded left-invertible weighted shift on a rootless directed tree admits a Wold-type decomposition. Approach: analyze the hyper-range via $g_{m, \mathfrak v}$ along bilateral paths, and derive necessary and sufficient conditions in terms of $\alpha_{\lambda}$ and $\alpha_{\lambda'}$, plus a balancedness criterion. Contributions: complete characterization; RKHS representations for the analytic part; construction of counterexamples showing limits of Wold-type decomposition in the rootless setting. Significance: clarifies the analytic/ wandering subspace structure of weighted shifts on directed trees and provides concrete models of operator-valued shifts with quantifiable conditions.

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.

Figures (2)

• Figure 1: The rootless directed tree $\mathcal{T}_{3,\infty}$
• Figure 2: The generations of $\mathcal{T}_{qb}$

Theorems & Definitions (37)

• Theorem 1.1
• Theorem 1.3
• Remark 1.4
• Lemma 2.1
• proof
• Remark 2.2
• Example 2.3
• Proposition 2.4
• proof
• Theorem 3.1