Wold decomposition for isometries with equal range

TL;DR

This paper establishes a unique Wold decomposition for n-tuples of isometries with equal range, providing analytic models and showing that wandering data serve as complete invariants, unifying previous results in the field.

Contribution

It introduces a novel Wold decomposition for isometries with equal range and demonstrates that wandering data are complete invariants, unifying prior work.

Findings

Unique Wold decomposition for isometries with equal range

Analytic models of the class of isometries

Wandering data are complete unitary invariants

Abstract

Let $n \geq 2$, and let $V=(V_1,\dots, V_n)$ be an $n$-tuple of isometries acting on a Hilbert space $\mathcal{H}$. We say that $V$ is an $n$-tuple of isometries with equal range if $V_i^{m_i}V_j^{m_j}\mathcal{H} = V_j^{m_j} V_i^{m_i}\mathcal{H}$ and $V_i^{*m_i}V_j^{m_j} \mathcal{H} = V_j^{m_j} V_i^{*m_i}\mathcal{H}$ for $m_i,m_j \in \mathbb{Z}_+$, where $1 \leq i<j \leq n$. We prove that each $n$-tuple of isometries with equal range admits a unique Wold decomposition. We obtain analytic models of the above class, and as a consequence, we show that the wandering data are complete unitary invariants for $n$-tuples of isometries with equal range. Our results unify all prior findings on the decomposition for tuples of isometries in the existing literature.