Distinguished varieties in the polydisc and dilation of commuting contractions

Abstract

A distinguished variety in the polydisc ${\mathbb{D}}^n$ is an affine complex algebraic variety that intersects ${\mathbb{D}}^n$ and exits the domain through the $n$-torus ${\mathbb{T}}^n$ without intersecting any other part of the topological boundary of ${\mathbb{D}}^n$. We find two different characterizations for a distinguished variety in the polydisc ${\mathbb{D}}^n$ in terms of the Taylor joint spectrum of certain linear matrix-pencils and thus generalize the seminal work due to Agler and M\raise.45ex\hbox{c}Carthy [Acta Math., 2005] on distinguished varieties in ${\mathbb{D}}^2$. We show that a distinguished variety in ${\mathbb{D}}^n$ is a part of an affine algebraic curve which is a set-theoretic complete intersection. We also show that if $(T_1,\dots ,T_n)$ is commuting tuple of Hilbert space contractions such that the defect space of $T={\prod }_{i=1}^n{T}_i$ is finite dimensional, then $(T_1,\dots ,T_n)$ admits a commuting unitary dilation $(U_1,\dots ,U_n)$ with $U={\prod }_{i=1}^n{U}_i$ being the minimal unitary dilation of $T$ if and only if some certain matrices associated with $(T_1,\dots ,T_n)$ define a distinguished variety in ${\mathbb{D}}^n$.

arXiv commentModified. new results added. 20 Pages