Rational dilation on the symmetrized tridisc: failure, success and unknown

Sourav Pal

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arXiv · 1610.00425

Rational dilation on the symmetrized tridisc: failure, success and unknown

Sourav Pal

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Abstract

The closed symmetrized tridisc $Γ_3$ and its distinguished boundary $bΓ_3$ are the sets $Γ_3=\{ (z_1+z_2+z_3,z_1z_2+z_2z_3+z_3z_1,z_1z_2z_3): \,|z_i|\leq 1, i=1,2,3 \}\subseteq \mathbb C^3$ $bΓ_3=\{ (z_1+z_2+z_3,z_1z_2+z_2z_3+z_3z_1,z_1z_2z_3): \,|z_i|= 1, i=1,2,3 \}\subseteq Γ_3.$ A triple of commuting operators $(S_1,S_2,P)$ defined on a Hilbert space $\mathcal H$ for which $Γ_3$ is a spectral set is called a $Γ_3$-contraction. In this article we show by a counter example that there are $Γ_3$-contractions which do not dilate. It is also shown that under certain conditions a $Γ_3$-contraction can have normal $bΓ_3$ dilation. We determine several classes of $Γ_3$-contractions which dilate and show explicit construction of their dilations. A concrete functional model is provided for the $Γ_3$-contractions which dilate. Various characterizations for $Γ_3$-unitaries and $Γ_3$-isometries are obtained; the classes of $Γ_3$-unitaries and $Γ_3$-isometries are analogous to the unitaries and isometries in one variable operator theory. Also we find out a model for the class of pure $Γ_3$-isometries. En route we study the geometry of the sets $Γ_3$ and $bΓ_3$ and provide variety of characterizations for them.

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BibTeX RIS

Sourav Pal. 2017-06-07. Rational dilation on the symmetrized tridisc: failure, success and unknown. https://arxiv.org/abs/1610.00425

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