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- W1926950215 abstract "We show an interplay between the complex geometry of the tetrablock $mathbb E$ and the commuting triples of operators having $overline{mathbb E}$ as a spectral set. We prove that every distinguished variety in the tetrablock is one-dimensional and can be represented as begin{equation}label{eqn:1} Omega={ (x_1,x_2,x_3)in mathbb E ,:, (x_1,x_2) in sigma_T(A_1^*+x_3A_2,,, A_2^*+x_3A_1) }, end{equation} where $A_1,A_2$ are commuting square matrices of the same order satisfying $[A_1^*,A_1]=[A_2^*,A_2]$ and a norm condition. The converse also holds, i.e, a set of the form (ref{eqn:1}) is always a distinguished variety in $mathbb E$. We show that for a triple of commuting operators $Upsilon = (T_1,T_2,T_3)$ having $overline{mathbb E}$ as a spectral set, there is a one-dimensional subvariety $Omega_{Upsilon}$ of $overline{mathbb E}$ depending on $Upsilon$ such that von-Neumann's inequality holds, i.e, [ f(T_1,T_2,T_3)leq sup_{(x_1,x_2,x_3)inOmega_{Upsilon}}, |f(x_1,x_2,x_3)|, ] for any holomorphic polynomial $f$ in three variables, provided that $T_3^nrightarrow 0$ strongly as $nrightarrow infty$. The variety $Omega_Upsilon$ has been shown to have representation like (ref{eqn:1}), where $A_1,A_2$ are the unique solutions of the operator equations begin{gather*} T_1-T_2^*T_3=(I-T_3^*T_3)^{frac{1}{2}}X_1(I-T_3^*T_3)^{frac{1}{2}} text{ and } T_2-T_1^*T_3=(I-T_3^*T_3)^{frac{1}{2}}X_2(I-T_3^*T_3)^{frac{1}{2}}. end{gather*} We also show that under certain condition, $Omega_{Upsilon}$ is a distinguished variety in $mathbb E$. We produce an explicit dilation and a concrete functional model for such a triple $(T_1,T_2,T_3)$ in which the unique operators $A_1,A_2$ play the main role. Also, we describe a connection of this theory with the distinguished varieties in the bidisc and in the symmetrized bidisc." @default.
- W1926950215 created "2016-06-24" @default.
- W1926950215 creator A5064136277 @default.
- W1926950215 date "2014-05-10" @default.
- W1926950215 modified "2023-09-27" @default.
- W1926950215 title "Subvarieties of the tetrablock and von-Neumann's inequality" @default.
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