Minimal isometric dilations and operator models for the polydisc
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| Format: | Preprint |
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2022
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| _version_ | 1866915034202898432 |
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| author | Pal, Sourav Sahasrabuddhe, Prajakta |
| author_facet | Pal, Sourav Sahasrabuddhe, Prajakta |
| contents | For commuting contractions $T_1,\dots ,T_n$ acting on a Hilbert space $\mathcal H$ with $T=\prod_{i=1}^n T_i$, we find a necessary and sufficient condition under which $(T_1,\dots ,T_n)$ dilates to commuting isometries $(V_1,\dots ,V_n)$ on the minimal isometric dilation space $T$, where $V=\prod_{i=1}^nV_i$ is the minimal isometric dilation of $T$. We construct both Sch$\ddot{a}$ffer and Sz. Nagy-Foias type isometric dilations for $(T_1,\dots ,T_n)$ on the minimal dilation spaces of $T$. Also, a different dilation is constructed when the product $T$ is a $C._0$ contraction, that is ${T^*}^n \rightarrow 0$ as $n \rightarrow \infty$. As a consequence of these dilation theorems we obtain different functional models for $(T_1,\dots ,T_n)$ in terms of multiplication operators on vectorial Hardy spaces. One notable fact about our models is that the multipliers are analytic functions in one variable. The dilation, when $T$ is a $C._0$ contraction, leads to a conditional factorization of a $T$. Several examples have been constructed. |
| format | Preprint |
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arxiv_https___arxiv_org_abs_2204_11391 |
| institution | arXiv |
| publishDate | 2022 |
| record_format | arxiv |
| spellingShingle | Minimal isometric dilations and operator models for the polydisc Pal, Sourav Sahasrabuddhe, Prajakta Functional Analysis Complex Variables For commuting contractions $T_1,\dots ,T_n$ acting on a Hilbert space $\mathcal H$ with $T=\prod_{i=1}^n T_i$, we find a necessary and sufficient condition under which $(T_1,\dots ,T_n)$ dilates to commuting isometries $(V_1,\dots ,V_n)$ on the minimal isometric dilation space $T$, where $V=\prod_{i=1}^nV_i$ is the minimal isometric dilation of $T$. We construct both Sch$\ddot{a}$ffer and Sz. Nagy-Foias type isometric dilations for $(T_1,\dots ,T_n)$ on the minimal dilation spaces of $T$. Also, a different dilation is constructed when the product $T$ is a $C._0$ contraction, that is ${T^*}^n \rightarrow 0$ as $n \rightarrow \infty$. As a consequence of these dilation theorems we obtain different functional models for $(T_1,\dots ,T_n)$ in terms of multiplication operators on vectorial Hardy spaces. One notable fact about our models is that the multipliers are analytic functions in one variable. The dilation, when $T$ is a $C._0$ contraction, leads to a conditional factorization of a $T$. Several examples have been constructed. |
| title | Minimal isometric dilations and operator models for the polydisc |
| topic | Functional Analysis Complex Variables |
| url | https://arxiv.org/abs/2204.11391 |