Wold decomposition for isometries with equal range
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arXiv
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| Auteurs principaux: | , |
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| Format: | Preprint |
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2023
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| _version_ | 1866917636414111744 |
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| author | Majee, Satyabrata Maji, Amit |
| author_facet | Majee, Satyabrata Maji, Amit |
| contents | 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. |
| format | Preprint |
| id |
arxiv_https___arxiv_org_abs_2309_04445 |
| institution | arXiv |
| publishDate | 2023 |
| record_format | arxiv |
| spellingShingle | Wold decomposition for isometries with equal range Majee, Satyabrata Maji, Amit Functional Analysis Complex Variables Operator Algebras 47A45, 47A15, 47A13, 47A05 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. |
| title | Wold decomposition for isometries with equal range |
| topic | Functional Analysis Complex Variables Operator Algebras 47A45, 47A15, 47A13, 47A05 |
| url | https://arxiv.org/abs/2309.04445 |