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| Format: | Preprint |
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2025
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| Online Access: | https://arxiv.org/abs/2511.00838 |
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| _version_ | 1866914130562121728 |
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| author | Pal, Avijit Paul, Bhaskar |
| author_facet | Pal, Avijit Paul, Bhaskar |
| contents | A fundamental theorem of Sz.-Nagy states that a contraction $T$ on a Hilbert space can be dilated to an isometry $V.$ A more multivariable context of recent significance for these concepts involves substituting the unit disk with $Γ_{E(3; 3; 1, 1, 1)}, Γ_{E(3; 2; 1, 2)},$ and pentablock. We demonstrate the necessary conditions for the existence of $Γ_{E(3; 3; 1, 1, 1)}$-isometric dilation, $Γ_{E(3; 2; 1, 2)}$-isometric dilation and pentablock-isometric dilation. We construct a class of $Γ_{E(3; 3; 1, 1, 1)}$-contractions and $Γ_{E(3; 2; 1, 2)}$-contractions that are always dilate . We create an example of a $Γ_{E(3; 3; 1, 1, 1)}$-contraction that has a $Γ_{E(3; 3; 1, 1, 1)}$-isometric dilation such that $[F_{7-i}^*, F_j] \ne [F_{7-j}^*, F_i] $ for some $i,j$ with $1\leq i ,j\leq 6,$ where $F_i$ and $F_{7-i}, 1\leq i \leq 6$ are the fundamental operators of $Γ_{E(3; 3; 1, 1, 1)}$-contraction $\textbf{T}=(T_1, \dots, T_7).$ We also produce an example of a $Γ_{E(3; 2; 1, 2)}$-contraction that has a $Γ_{E(3; 2; 1, 2)}$-isometric dilation by which $$[G^*_1, G_1] \neq [\tilde{G}^*_2, \tilde{G}_2]~{\rm{ and }}~[2G^*_2, 2G_2] \neq
[2\tilde{G}^*_1, 2\tilde{G}_1],$$ where $G_1, 2G_2, 2\tilde{G}_1, \tilde{G}_2$ are the fundamental operators of $\textbf{S}$. As a result, the set of sufficient conditions for the existence of a $Γ_{E(3; 3; 1, 1, 1)}$-isometric dilation and $Γ_{E(3; 2; 1; 2)} $-isometric dilations presented in Theorem \ref{conddilation} and Theorem \ref{condilation1}, respectively, are not generally necessary. We construct explicit $Γ_{E(3; 3; 1, 1, 1)} $-isometric, $Γ_{E(3; 2; 1; 2)} $-isometric dilations and $\mathcal{\bar{P}}$-isometric dilation of $Γ_{E(3; 3; 1, 1, 1)}$-contraction, $Γ_{E(3; 2; 1; 2)}$-contraction and $\mathcal{\bar{P}}$-contraction, respectively. |
| format | Preprint |
| id |
arxiv_https___arxiv_org_abs_2511_00838 |
| institution | arXiv |
| publishDate | 2025 |
| record_format | arxiv |
| spellingShingle | Necessary Conditions for $Γ_{E(3; 3; 1, 1, 1)}$-Isometric Dilation, $Γ_{E(3; 2; 1, 2)}$-Isometric Dilation and $\mathcal{\bar{P}}$-Isometric Dilation Pal, Avijit Paul, Bhaskar Functional Analysis A fundamental theorem of Sz.-Nagy states that a contraction $T$ on a Hilbert space can be dilated to an isometry $V.$ A more multivariable context of recent significance for these concepts involves substituting the unit disk with $Γ_{E(3; 3; 1, 1, 1)}, Γ_{E(3; 2; 1, 2)},$ and pentablock. We demonstrate the necessary conditions for the existence of $Γ_{E(3; 3; 1, 1, 1)}$-isometric dilation, $Γ_{E(3; 2; 1, 2)}$-isometric dilation and pentablock-isometric dilation. We construct a class of $Γ_{E(3; 3; 1, 1, 1)}$-contractions and $Γ_{E(3; 2; 1, 2)}$-contractions that are always dilate . We create an example of a $Γ_{E(3; 3; 1, 1, 1)}$-contraction that has a $Γ_{E(3; 3; 1, 1, 1)}$-isometric dilation such that $[F_{7-i}^*, F_j] \ne [F_{7-j}^*, F_i] $ for some $i,j$ with $1\leq i ,j\leq 6,$ where $F_i$ and $F_{7-i}, 1\leq i \leq 6$ are the fundamental operators of $Γ_{E(3; 3; 1, 1, 1)}$-contraction $\textbf{T}=(T_1, \dots, T_7).$ We also produce an example of a $Γ_{E(3; 2; 1, 2)}$-contraction that has a $Γ_{E(3; 2; 1, 2)}$-isometric dilation by which $$[G^*_1, G_1] \neq [\tilde{G}^*_2, \tilde{G}_2]~{\rm{ and }}~[2G^*_2, 2G_2] \neq [2\tilde{G}^*_1, 2\tilde{G}_1],$$ where $G_1, 2G_2, 2\tilde{G}_1, \tilde{G}_2$ are the fundamental operators of $\textbf{S}$. As a result, the set of sufficient conditions for the existence of a $Γ_{E(3; 3; 1, 1, 1)}$-isometric dilation and $Γ_{E(3; 2; 1; 2)} $-isometric dilations presented in Theorem \ref{conddilation} and Theorem \ref{condilation1}, respectively, are not generally necessary. We construct explicit $Γ_{E(3; 3; 1, 1, 1)} $-isometric, $Γ_{E(3; 2; 1; 2)} $-isometric dilations and $\mathcal{\bar{P}}$-isometric dilation of $Γ_{E(3; 3; 1, 1, 1)}$-contraction, $Γ_{E(3; 2; 1; 2)}$-contraction and $\mathcal{\bar{P}}$-contraction, respectively. |
| title | Necessary Conditions for $Γ_{E(3; 3; 1, 1, 1)}$-Isometric Dilation, $Γ_{E(3; 2; 1, 2)}$-Isometric Dilation and $\mathcal{\bar{P}}$-Isometric Dilation |
| topic | Functional Analysis |
| url | https://arxiv.org/abs/2511.00838 |