The toral contractions and $\Gamma$-distinguished $\Gamma$-contractions

A pair of commuting Hilbert space contractions $(T_1,T_2)$ is said to be toral if there is a polynomial $p \in \mathbb C[z_1,z_2]$ such that its zero set $Z(p)$ defines a distinguished variety in the bidisc $\mathbb D^2$ and $p(T_1,T_2)=0$. A pair of commuting Hilbert space operators $(S,P)$ is said to be a $\Gamma$-contraction if the closed symmetrized bidisc $$\Gamma=\{ (z_1+z_2,z_1z_2)\,:\, |z_1|, \, |z_2| \leq 1 \}$$ is a spectral set for $(S,P)$. A $\Gamma$-contraction $(S,P)$ is called $\Gamma$-distinguished if $q(S,P)=0$ for some polynomial $q\in \mathbb C[z_1,z_2]$ whose zero set $Z(q)$ gives rise to a distinguished variety in the symmetrized bidisc $\mathbb G_2$. We find necessary and sufficient conditions such that a toral pair of contractions dilates to a toral pair of isometries. In the same spirit, we characterize all $\Gamma$-distinguished $\Gamma$-contractions that admit dilation to $\Gamma$-distinguished $\Gamma$-isometries. The distinguished boundary of a distinguished variety in $\mathbb D^2$ and $\mathbb G_2$ is determined. Examples are provided at places to show the contrasts between the theory of toral contractions and $\Gamma$-distinguished $\Gamma$-contractions.