Papers
Topics
Authors
Recent
Gemini 2.5 Flash
Gemini 2.5 Flash
143 tokens/sec
GPT-4o
7 tokens/sec
Gemini 2.5 Pro Pro
46 tokens/sec
o3 Pro
4 tokens/sec
GPT-4.1 Pro
38 tokens/sec
DeepSeek R1 via Azure Pro
28 tokens/sec
2000 character limit reached

Distinguished varieties in a family of domains associated with spectral interpolation and operator theory (2010.16077v4)

Published 30 Oct 2020 in math.FA, math.AG, and math.CV

Abstract: We find characterization for the distinguished varieties in the symmetrized polydisc $\mathbb G_n \; (n\geq 2)$ 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 $(S_1*, \dots , S_{n-1},P^)$ 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 $(S_1*, \dots ,S_{n-1},P^)$ is minimal and acts on the minimal unitary dilation space of $P*$. Also, we show interplay between the distinguished varieties in $\mathbb G_2$ and $\mathbb G_{3}$.

Summary

We haven't generated a summary for this paper yet.