Necessary Conditions for $\Gamma_{E(3; 3; 1, 1, 1)}$-Isometric Dilation, $\Gamma_{E(3; 2; 1, 2)}$-Isometric Dilation and $\mathcal{\bar{P}}$-Isometric Dilation

TL;DR

This paper investigates the conditions under which certain multivariable operator contractions can be dilated to isometries within complex geometric domains, providing explicit examples and analyzing the necessity of these conditions.

Contribution

It establishes necessary conditions for specific multivariable isometric dilations and constructs explicit dilation examples, challenging the sufficiency of previously known conditions.

Findings

Necessary conditions for $ ext{Gamma}$-isometric dilations are identified.

Explicit examples of dilations are constructed.

Sufficient conditions are shown not to be necessary in general.

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 $\Gamma_{E(3; 3; 1, 1, 1)}, \Gamma_{E(3; 2; 1, 2)},$ and pentablock. We demonstrate the necessary conditions for the existence of $\Gamma_{E(3; 3; 1, 1, 1)}$-isometric dilation, $\Gamma_{E(3; 2; 1, 2)}$-isometric dilation and pentablock-isometric dilation. We construct a class of $\Gamma_{E(3; 3; 1, 1, 1)}$-contractions and $\Gamma_{E(3; 2; 1, 2)}$-contractions that are always dilate . We create an example of a $\Gamma_{E(3; 3; 1, 1, 1)}$-contraction that has a $\Gamma_{E(3; 3; 1, 1, 1)}$-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 $\Gamma_{E(3; 3; 1, 1, 1)}$-contraction $\textbf{T}=(T_1, \dots, T_7).$ We also produce an example of a $\Gamma_{E(3; 2; 1, 2)}$-contraction that has a $\Gamma_{E(3; 2; 1, 2)}$-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 $\Gamma_{E(3; 3; 1, 1, 1)}$-isometric dilation and $\Gamma_{E(3; 2; 1; 2)} $-isometric dilations presented in Theorem \ref{conddilation} and Theorem \ref{condilation1}, respectively, are not generally necessary. We construct explicit $\Gamma_{E(3; 3; 1, 1, 1)} $-isometric, $\Gamma_{E(3; 2; 1; 2)} $-isometric dilations and $\mathcal{\bar{P}}$-isometric dilation of $\Gamma_{E(3; 3; 1, 1, 1)}$-contraction, $\Gamma_{E(3; 2; 1; 2)}$-contraction and $\mathcal{\bar{P}}$-contraction, respectively.