The toral contractions and $Γ$-distinguished $Γ$-contractions
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arXiv
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| Natura: | Preprint |
| Pubblicazione: |
2022
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| _version_ | 1866908518549815296 |
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| author | Pal, Sourav Tomar, Nitin |
| author_facet | Pal, Sourav Tomar, Nitin |
| contents | A pair of commuting Hilbert space contractions $(T_1,T_2)$ is said to be toral if there is a polynomial $p \in \mathbb C[z_1,z_2]$ such that its zero set $Z(p)$ defines a distinguished variety in the bidisc $\mathbb D^2$ and $p(T_1,T_2)=0$. A pair of commuting Hilbert space operators $(S,P)$ is said to be a $Γ$-contraction if the closed symmetrized bidisc \[ Γ=\{ (z_1+z_2,z_1z_2)\,:\, |z_1|, \, |z_2| \leq 1 \} \] is a spectral set for $(S,P)$. A $Γ$-contraction $(S,P)$ is called $Γ$-distinguished if $q(S,P)=0$ for some polynomial $q\in \mathbb C[z_1,z_2]$ whose zero set $Z(q)$ gives rise to a distinguished variety in the symmetrized bidisc $\mathbb G_2$. We find necessary and sufficient conditions such that a toral pair of contractions dilates to a toral pair of isometries. In the same spirit, we characterize all $Γ$-distinguished $Γ$-contractions that admit dilation to $Γ$-distinguished $Γ$-isometries. The distinguished boundary of a distinguished variety in $\mathbb D^2$ and $\mathbb G_2$ is determined. Examples are provided at places to show the contrasts between the theory of toral contractions and $Γ$-distinguished $Γ$-contractions. |
| format | Preprint |
| id |
arxiv_https___arxiv_org_abs_2204_08380 |
| institution | arXiv |
| publishDate | 2022 |
| record_format | arxiv |
| spellingShingle | The toral contractions and $Γ$-distinguished $Γ$-contractions Pal, Sourav Tomar, Nitin Functional Analysis Complex Variables A pair of commuting Hilbert space contractions $(T_1,T_2)$ is said to be toral if there is a polynomial $p \in \mathbb C[z_1,z_2]$ such that its zero set $Z(p)$ defines a distinguished variety in the bidisc $\mathbb D^2$ and $p(T_1,T_2)=0$. A pair of commuting Hilbert space operators $(S,P)$ is said to be a $Γ$-contraction if the closed symmetrized bidisc \[ Γ=\{ (z_1+z_2,z_1z_2)\,:\, |z_1|, \, |z_2| \leq 1 \} \] is a spectral set for $(S,P)$. A $Γ$-contraction $(S,P)$ is called $Γ$-distinguished if $q(S,P)=0$ for some polynomial $q\in \mathbb C[z_1,z_2]$ whose zero set $Z(q)$ gives rise to a distinguished variety in the symmetrized bidisc $\mathbb G_2$. We find necessary and sufficient conditions such that a toral pair of contractions dilates to a toral pair of isometries. In the same spirit, we characterize all $Γ$-distinguished $Γ$-contractions that admit dilation to $Γ$-distinguished $Γ$-isometries. The distinguished boundary of a distinguished variety in $\mathbb D^2$ and $\mathbb G_2$ is determined. Examples are provided at places to show the contrasts between the theory of toral contractions and $Γ$-distinguished $Γ$-contractions. |
| title | The toral contractions and $Γ$-distinguished $Γ$-contractions |
| topic | Functional Analysis Complex Variables |
| url | https://arxiv.org/abs/2204.08380 |