Constrained dilation and $\hat{I}“$-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{$\hat{I}“$-contraction}. A $\hat{I}“$-contraction $(S,P)$ is called \textit{$\hat{I}“$-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 $\hat{I}“$-isometric dilation of a $\hat{I}“$-contraction $(S,P)$ in the literature. In this article, we study when such a minimal $\hat{I}“$-isometric dilation is $\hat{I}“$-distinguished provided that $(S,P)$ is a $\hat{I}“$-distinguished $\hat{I}“$-contraction. We show that a pure $\hat{I}“$-isometry $(T,V)$ with defect space $\dim {\mathcal{D}}_{V^{\ast }}<\infty$, is $\hat{I}“$-distinguished if and only if the fundamental operator of $(T^{\ast },V^{\ast })$ has numerical radius less than $1$. Further, it is proved that a $\hat{I}“$-contraction acting on a finite-dimensional Hilbert space dilates to a $\hat{I}“$-distinguished $\hat{I}“$-isometry if its fundamental operator has numerical radius less than $1$. We also provide sufficient conditions for a pure $\hat{I}“$-contraction to be $\hat{I}“$-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 $\hat{I}“$-distinguished $\hat{I}“$-unitaries and $\hat{I}“$-distinguished pure $\hat{I}“$-isometries.

23 pages, Submitted to Journal