Minimal unitary dilations for commuting contractions
Fuente:
arXiv
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| Natura: | Preprint |
| Pubblicazione: |
2022
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| _version_ | 1866917570280423424 |
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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 show that $(T_1, \dots, T_n)$ dilates to commuting isometries $(V_1, \dots , V_n)$ on the minimal isometric dilation space of $T$ with $V=\prod_{i=1}^n V_i$ being the minimal isometric dilation of $T$ if and only if $(T_1^*, \dots , T_n^*)$ dilates to commuting isometries $(Y_1, \dots , Y_n)$ on the minimal isometric dilation space of $T^*$ with $Y=\prod_{i=1}^n Y_i$ being the minimal isometric dilation of $T^*$. Then, we prove an analogue of this result for unitary dilations of $(T_1, \dots , T_n)$ and its adjoint. We find a necessary and sufficient condition such that $(T_1, \dots , T_n)$ possesses a unitary dilation $(W_1, \dots , W_n)$ on the minimal unitary dilation space of $T$ with $W=\prod_{i=1}^n W_i$ being the minimal unitary dilation of $T$. We show an explicit construction of such a unitary dilation on both Sch$\ddot{a}$ffer and Sz. Nagy-Foias minimal unitary dilation spaces of $T$. Also, we show that a relatively weaker hypothesis is necessary and sufficient for the existence of such a unitary dilation when $T$ is a $C._0$ contraction, i.e. when ${T^*}^n \rightarrow 0$ strongly as $n \rightarrow \infty $. We construct a different unitary dilation for $(T_1, \dots , T_n)$ when $T$ is a $C._0$ contraction. |
| format | Preprint |
| id |
arxiv_https___arxiv_org_abs_2205_09093 |
| institution | arXiv |
| publishDate | 2022 |
| record_format | arxiv |
| spellingShingle | Minimal unitary dilations for commuting contractions Pal, Sourav Sahasrabuddhe, Prajakta Functional Analysis Operator Algebras For commuting contractions $T_1,\dots ,T_n$ acting on a Hilbert space $\mathcal H$ with $T=\prod_{i=1}^n T_i$, we show that $(T_1, \dots, T_n)$ dilates to commuting isometries $(V_1, \dots , V_n)$ on the minimal isometric dilation space of $T$ with $V=\prod_{i=1}^n V_i$ being the minimal isometric dilation of $T$ if and only if $(T_1^*, \dots , T_n^*)$ dilates to commuting isometries $(Y_1, \dots , Y_n)$ on the minimal isometric dilation space of $T^*$ with $Y=\prod_{i=1}^n Y_i$ being the minimal isometric dilation of $T^*$. Then, we prove an analogue of this result for unitary dilations of $(T_1, \dots , T_n)$ and its adjoint. We find a necessary and sufficient condition such that $(T_1, \dots , T_n)$ possesses a unitary dilation $(W_1, \dots , W_n)$ on the minimal unitary dilation space of $T$ with $W=\prod_{i=1}^n W_i$ being the minimal unitary dilation of $T$. We show an explicit construction of such a unitary dilation on both Sch$\ddot{a}$ffer and Sz. Nagy-Foias minimal unitary dilation spaces of $T$. Also, we show that a relatively weaker hypothesis is necessary and sufficient for the existence of such a unitary dilation when $T$ is a $C._0$ contraction, i.e. when ${T^*}^n \rightarrow 0$ strongly as $n \rightarrow \infty $. We construct a different unitary dilation for $(T_1, \dots , T_n)$ when $T$ is a $C._0$ contraction. |
| title | Minimal unitary dilations for commuting contractions |
| topic | Functional Analysis Operator Algebras |
| url | https://arxiv.org/abs/2205.09093 |