On the Dilation Theory and Canonical Decomposition of $\mathbf{\Theta}_n$-Contractions

Abstract

This paper studies the domain $\mathbf{\Theta}_n$ from the perspective of operator theory. We obtain several characterizations of $\mathbf{\Theta}_n$-contractions (respectively, $\mathbf{\Theta}_n$-unitaries and $\mathbf{\Theta}_n$-isometries) and establish their relationships with $\Gamma_n$-contractions (respectively, $\Gamma_n$-unitaries and $\Gamma_n$-isometries), tetrablock contractions (respectively, tetrablock unitaries and tetrablock isometries), and $\mathbf{\Theta}_{n+1}$-contractions (respectively, $\mathbf{\Theta}_{n+1}$-unitaries and $\mathbf{\Theta}_{n+1}$-isometries). We prove that every $\mathbf{\Theta}_n$-contraction admits a canonical decomposition into the direct sum of a $\mathbf{\Theta}_n$-unitary and a completely non-unitary $\mathbf{\Theta}_n$-contraction. We further develop a dilation theory for $\mathbf{\Theta}_n$-contractions by obtaining necessary and sufficient conditions for the existence of minimal $\mathbf{\Theta}_n$-isometric dilations. As an application, we show that the minimal $\Gamma_n$-isometric dilation arises as a special case of the minimal $\mathbf{\Theta}_n$-isometric dilation. Finally, we identify a class of $\mathbf{\Theta}_2$-contractions that always admit $\mathbf{\Theta}_2$-isometric extensions.