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.

This is the initial version. Final version will submit soon