Wold decomposition for isometries with equal range

Abstract

Let $n\ge 2$, 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 $V_i^{\ast m_i}{V}_j^{m_j}\mathcal{H}=V_j^{m_j}{V}_i^{\ast m_i}\mathcal{H}$ for $m_i,m_j\in {\mathbb{Z}}_{+}$, where $1\le i<j\le n$. 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.

arXiv comment33 pages. Sections 6 and 7 added in the revised version