Distinguished varieties in a family of domains associated with spectral interpolation and operator theory

Abstract

We find characterization for the distinguished varieties in the symmetrized polydisc ${\mathbb{G}}_n(n\ge 2)$ and thus generalize the work [\textit{J. Funct. Anal.}, 266 (2014), 5779 -- 5800] on ${\mathbb{G}}_2$ by the author and Shalit. We show that a distinguished variety $\Lambda$ in ${\mathbb{G}}_n$ is a part of an algebraic curve, which is a set-theoretic complete intersection, and that $\Lambda$ can be represented by the Taylor joint spectrum of $n-1$ commuting scalar matrices satisfying certain conditions. An $n$-tuple of commuting Hilbert space operators $(S_1,\dots ,S_{n-1},P)$ for which ${\Gamma }_n=\overline{{\mathbb{G}}_n}$ is a spectral set is called a ${\Gamma }_n$-contraction. To every ${\Gamma }_n$-contraction $(S_1,\dots ,S_{n-1},P)$ there is a unique operator tuple $(F_1,\dots ,F_{n-1})$, called the ${\mathcal{F}}_O$-tuple of $(S_1,\dots ,S_{n-1},P)$, satisfying \[ S_i-S_{n-i}^*P=D_PF_iD_P \,,\quad i=1, \dots ,n-1. \] We produce concrete functional model for the pure isometric-operator tuples associated with ${\Gamma }_n$ and by an application of that model we establish that the ${\Gamma }_n$-contractions $(S_1,\dots ,S_{n-1},P)$ and $(S_1^{\ast },\dots ,S_{n-1}^{\ast },P^{\ast })$ admit normal $\partial {\overline{\Lambda }}_{\Sigma }-$dilations for a unique distinguished variety ${\Lambda }_{\Sigma }$ in ${\mathbb{G}}_n$, when ${\Lambda }_{\Sigma }$ is determined by the ${\mathcal{F}}_O$-tuple of $(S_1,\dots ,S_{n-1},P)$. Further, we show that the dilation of $(S_1^{\ast },\dots ,S_{n-1}^{\ast },P^{\ast })$ is minimal and acts on the minimal unitary dilation space of $P^{\ast }$. Also, we show interplay between the distinguished varieties in ${\mathbb{G}}_2$ and ${\mathbb{G}}_3$.

arXiv commentAnnali della Scuola Normale Superiore di Pisa. Classe di Scienze, To appear