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
Avijit Pal, Bhaskar Paul
TL;DR
This work advances multivariable dilation theory for Γ_E domains by identifying necessary conditions for Γ_{E(3;3;1,1,1)}- and Γ_{E(3;2;1,2)}-isometric dilations as well as for 𝔓̄-contractions to have 𝔓̄-isometric dilations. It connects the fundamental operator framework with dilations, showing that previously stated sufficient conditions are not generally necessary, and provides explicit dilation constructions for specific contraction classes. The paper also proves key structural lemmas when top coordinates are partial isometries and develops concrete examples and families of dilations, including 𝔓̄-dilations, thereby enriching the landscape of spectral-set based dilations in the pentablock/tetrablock context. Overall, it highlights both limitations of existing sufficient criteria and new explicit dilation models with potential applications in multivariable operator theory and complex geometry of these domains.
Abstract
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.