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.
Categories: math.FA