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- W2774832789 abstract "The open and closed textit{symmetrized polydisc} or, textit{symmetrized $n$-disc} for $ngeq 2$, are the following subsets of $mathbb C^n$: begin{align*} mathbb G_n &=left{ left(sum_{1leq ileq n} z_i,sum_{1leq i<jleq n}z_iz_j,dots, prod_{i=1}^n z_i right): ,|z_i|< 1, i=1,dots,n right }, Gamma_n & =left{ left(sum_{1leq ileq n} z_i,sum_{1leq i<jleq n}z_iz_j,dots, prod_{i=1}^n z_i right): ,|z_i|leq 1, i=1,dots,n right }. end{align*} A tuple of commuting $n$ operators $(S_1,dots,S_{n-1},P)$ defined on a Hilbert space $mathcal H$ for which $Gamma_n$ is a spectral set is called a $Gamma_n$-contraction. In this article, we show by a counter example that rational dilation fails on the symmetrized $n$-disc for any $ngeq 3$. We find new characterizations for the points in $mathbb G_n$ and $Gamma_n$. We also present few new characterizations for the $Gamma_n$-unitaries and $Gamma_n$-isometries." @default.
- W2774832789 created "2017-12-22" @default.
- W2774832789 creator A5064136277 @default.
- W2774832789 date "2017-12-14" @default.
- W2774832789 modified "2023-09-27" @default.
- W2774832789 title "The failure of rational dilation on the symmetrized $n$-disk for any $ngeq 3$" @default.
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