Wold decomposition for isometries with equal range

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.