Constrained dilation and $\Gamma$-contractions

A commuting pair of Hilbert space operators having the closed symmetrized bidisc $$\Gamma=\{(z_1+z_2, z_1z_2) \in \mathbb C^2 \ : \ |z_1| \leq 1, |z_2| \leq 1\}$$ as a spectral set is called a \textit{$\Gamma$-contraction}. A $\Gamma$-contraction $(S,P)$ is called \textit{$\Gamma$-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 $\Gamma$-isometric dilation of a $\Gamma$-contraction $(S,P)$ in the literature. In this article, we study when such a minimal $\Gamma$-isometric dilation is $\Gamma$-distinguished provided that $(S,P)$ is a $\Gamma$-distinguished $\Gamma$-contraction. We show that a pure $\Gamma$-isometry $(T,V)$ with defect space $\dim \mathcal D_{V^*}< \infty$, is $\Gamma$-distinguished if and only if the fundamental operator of $(T^*,V^*)$ has numerical radius less than $1$. Further, it is proved that a $\Gamma$-contraction acting on a finite-dimensional Hilbert space dilates to a $\Gamma$-distinguished $\Gamma$-isometry if its fundamental operator has numerical radius less than $1$. We also provide sufficient conditions for a pure $\Gamma$-contraction to be $\Gamma$-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 $\Gamma$-distinguished $\Gamma$-unitaries and $\Gamma$-distinguished pure $\Gamma$-isometries.