Journal of Operator Theory

Volume 95, Issue 2, Spring 2026 pp. 305-337.

Wold decomposition for isometries with equal range

Authors: Satyabrata Majee (1), Amit Maji (2)
Author institution: (1) Department of Mathematics, Indian Institute of Technology Roorkee, Roorkee, Uttarakhand, 247667, India
(2) Indian Institute of Technology Roorkee, Department of Mathematics, Roorkee-247667, Uttarakhand, India

Summary: Let $n \geqslant 2$, 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 {\textit{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 $1 \leqslant i$\textless$j \leqslant n$. We prove that each $n$-tuple of {\textit{isometries with equal range}} admits a unique {\textit{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 {\textit{isometries with equal range}}. Our results unify all prior findings on the decomposition for tuples of isometries in the existing literature.

DOI: http://dx.doi.org/10.7900/jot.2023sep07.2501
Keywords: isometries, Wold--von Neumann decomposition, shift operators, unitary operators, invariant subspaces

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