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- W2528610080 abstract "We show that every distinguished variety in the symmetrized tridisc $mathbb G_3$ is one-dimensional and can be represented as begin{equation}label{eqn:1} Lambda={ (s_1,s_2,p)in mathbb G_3 ,:, (s_1,s_2) in sigma_T(F_1^*+pF_2,,, F_2^*+pF_1) }, end{equation} where $F_1,F_2$ are commuting square matrices of the same order satisfying $[F_1^*,F_1]=[F_2^*,F_2]$ and a norm condition. The converse also holds, i.e, a set of the form (ref{eqn:1}) is always a distinguished variety in $mathbb G_3$. We show that for a triple of commuting operators $Sigma = (S_1,S_2,P)$ having $Gamma_3$ as a spectral set, there is a one-dimensional subvariety $Lambda_{Sigma}$ of $Gamma_3$ depending on $Sigma$ such that von-Neumann's inequality holds, i.e, [ f(S_1,S_2,P)leq sup_{(s_1,s_2,p)inLambda_{Sigma}}, |f(s_1,s_2,p)|, ] for any holomorphic polynomial $f$ in three 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 $F_1,F_2$ are the unique solutions of the operator equations begin{gather*} S_1-S_2^*P=(I-P^*P)^{frac{1}{2}}X_1(I-P^*P)^{frac{1}{2}} text{ and } S_2-S_1^*P=(I-P^*P)^{frac{1}{2}}X_2(I-P^*P)^{frac{1}{2}}. end{gather*} We also show that under certain condition, $Lambda_Sigma$ is a distinguished variety in $mathbb G_3$. We produce an explicit dilation and a concrete functional model for such a triple $(S_1,S_2,P)$ in which the unique operators $F_1,F_2$ play the main role. Also, we describe a connection of this theory with the distinguished varieties in the symmetrized bidisc and in the unit bidisc $mathbb D^2$." @default.
- W2528610080 created "2016-10-14" @default.
- W2528610080 creator A5064136277 @default.
- W2528610080 date "2016-10-04" @default.
- W2528610080 modified "2023-09-27" @default.
- W2528610080 title "Operator theory and representation of distinguished varieties in the symmetrized tridisc" @default.
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