On the Dilation Theory and Canonical Decomposition of $\mathbfΘ_n$-Contractions

Abstract

This paper studies the domain $\mathbfΘ_n$ from the perspective of operator theory. We obtain several characterizations of $\mathbfΘ_n$-contractions (respectively, $\mathbfΘ_n$-unitaries and $\mathbfΘ_n$-isometries) and establish their relationships with $Γ_n$-contractions (respectively, $Γ_n$-unitaries and $Γ_n$-isometries), tetrablock contractions (respectively, tetrablock unitaries and tetrablock isometries), and $\mathbfΘ_{n+1}$-contractions (respectively, $\mathbfΘ_{n+1}$-unitaries and $\mathbfΘ_{n+1}$-isometries). We prove that every $\mathbfΘ_n$-contraction admits a canonical decomposition into the direct sum of a $\mathbfΘ_n$-unitary and a completely non-unitary $\mathbfΘ_n$-contraction. We further develop a dilation theory for $\mathbfΘ_n$-contractions by obtaining necessary and sufficient conditions for the existence of minimal $\mathbfΘ_n$-isometric dilations. As an application, we show that the minimal $Γ_n$-isometric dilation arises as a special case of the minimal $\mathbfΘ_n$-isometric dilation. Finally, we identify a class of $\mathbfΘ_2$-contractions that always admit $\mathbfΘ_2$-isometric extensions.