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- W2749042607 abstract "We prove that every distinguished variety in the symmetrized polydisc $mathbb G_n$ has complex dimension $1$ and can be represented as begin{align}label{eqn:1} Lambda= &{ (s_1,dots,s_{n-1},p)in mathbb G_n ,: nonumber & quad (s_1,dots,s_{n-1}) in sigma_T(F_1^*+pF_{n-1},,, F_2^*+pF_{n-2},,,dots,, F_{n-1}^*+pF_1) }, end{align} where $F_1,dots,F_{n-1}$ are commuting square matrices of same order satisfying begin{itemize} item[(i)] $[F_i^*,F_{n-j}]=[F_j^*,F_{n-i}]$ for $1leq i<jleq n-1$, item[(ii)] $sigma_T(F_1,dots,F_{n-1})subseteq mathbb G_{n-1}$. end{itemize} The converse also holds, i.e, a set of the form (ref{eqn:1}) is always a distinguished variety in $mathbb G_n$. We show that for a tuple of commuting operators $Sigma = (S_1,dots,S_{n-1},P)$ having $Gamma_n$ as a spectral set, there is a distinguished variety $Lambda_{Sigma}$ in $mathbb G_n$ such that the von-Neumann's inequality holds on $overline{Lambda_{Sigma}}$, i.e, [ |f(S_1,dots,S_{n-1},P)|leq sup_{(s_1,dots,s_{n-1},p)inLambda_{Sigma}}, |f(s_1,dots,s_{n-1},p)|, ] for any holomorphic polynomial $f$ in $n$ variables, provided that $P^nrightarrow 0$ strongly as $nrightarrow infty$. The variety $Lambda_Sigma$ has been shown to have representation like (ref{eqn:1}), where $F_i$ is the unique solutions of the operator equation [ S_i-S_{n-i}^*P=(I-P^*P)^{frac{1}{2}}X_i(I-P^*P)^{frac{1}{2}},,,i=1,dots,n-1. ] We provide some operator theory on $Gamma_n$. We produce an explicit dilation and a concrete functional model for such a triple $(S_1,dots,S_{n-1},P)$ and the unique operators $F_1,dots,F_{n-1}$ play central role in this model. Also for $ngeq 3$, we describe a connection between distinguished varieties in $mathbb G_n$ and $mathbb G_{n-1}$." @default.
- W2749042607 created "2017-08-31" @default.
- W2749042607 creator A5064136277 @default.
- W2749042607 date "2017-08-20" @default.
- W2749042607 modified "2023-09-27" @default.
- W2749042607 title "Operator theory and distinguished varieties in the symmetrized $n$-disk" @default.
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