Constrained dilation and $Γ$-contractions

Abstract

A commuting pair of Hilbert space operators having the closed symmetrized bidisc \[ Γ=\{(z_1+z_2, z_1z_2) \in \mathbb C^2 \ : \ |z_1| \leq 1, |z_2| \leq 1\} \] as a spectral set is called a \textit{$Γ$-contraction}. A $Γ$-contraction $(S,P)$ is called \textit{$Γ$-distinguished} if $(S,P)$ is annihilated by a polynomial $q \in \mathbb C[z_1,z_2]$ whose zero set $Z(q)$ defines a distinguished variety in the symmetrized bidisc $\mathbb G$. There is Schaffer-type minimal $Γ$-isometric dilation of a $Γ$-contraction $(S,P)$ in the literature. In this article, we study when such a minimal $Γ$-isometric dilation is $Γ$-distinguished provided that $(S,P)$ is a $Γ$-distinguished $Γ$-contraction. We show that a pure $Γ$-isometry $(T,V)$ with defect space $\dim \mathcal D_{V^*}< \infty$, is $Γ$-distinguished if and only if the fundamental operator of $(T^*,V^*)$ has numerical radius less than $1$. Further, it is proved that a $Γ$-contraction acting on a finite-dimensional Hilbert space dilates to a $Γ$-distinguished $Γ$-isometry if its fundamental operator has numerical radius less than $1$. We also provide sufficient conditions for a pure $Γ$-contraction to be $Γ$-distinguished. Wold decomposition splits an isometry into two orthogonal parts of which one is a unitary and the other is a completely non-unitary contraction. In this direction, we find a few decomposition results for the $Γ$-distinguished $Γ$-unitaries and $Γ$-distinguished pure $Γ$-isometries.