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

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 $[F_{7-i}^{\ast },F_j]\ne [F_{7-j}^{\ast },F_i]$ for some $i,j$ with $1\le i,j\le 6,$ where $F_i$ and $F_{7-i},1\le i\le 6$ are the fundamental operators of ${\Gamma }_{E(3;3;1,1,1)}$-contraction $\text{T}=(T_1,\dots ,T_7).$ 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 $$[G_1^{\ast },G_1]\ne [{\tilde{G}}_2^{\ast },{\tilde{G}}_2]\mathrm{and}[2G_2^{\ast },2G_2]\ne [2{\tilde{G}}_1^{\ast },2{\tilde{G}}_1],$$ where $G_1,2G_2,2{\tilde{G}}_1,{\tilde{G}}_2$ are the fundamental operators of $\text{S}$. 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 $\bar{\mathcal{P}}$-isometric dilation of ${\Gamma }_{E(3;3;1,1,1)}$-contraction, ${\Gamma }_{E(3;2;1;2)}$-contraction and $\bar{\mathcal{P}}$-contraction, respectively.