Rational dilation on the polydisc and distinguished varieties

Abstract:  We define spectral set, rational dilation and recall a brief literature of success of rational dilation on the unit disk $\mathbb D= \{  z\in \mathbb C : |z|<1 \}$, on an annulus, on the bidisk $\mathbb D^2$, symmetrized bidisc $\mathbb G_2$ and its failure on a triply connected domain and on the polydisk $\mathbb D^n$ for $n \geq 3$. We find a necessary and sufficient condition such that a tuple of commuting Hilbert space contractions $(T_1, \dots , T_n)$ having the closed polydisk $\mathbb D^n$ ($n>2$) as a spectral set admits a normal distinguished-boundary dilation on the minimal dilation space of the product $\prod_{i=1}^n T_i$ and show an explicit construction of such a dilation. We also show interaction of such a dilation with a class of algebraic curves called distinguished varieties when $T_1, \dots , T_n$ are commuting matrices.