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

Abstract

The closed symmetrized tridisc ${\Gamma }_3$ and its distinguished boundary $b{\Gamma }_3$ are the sets ${\Gamma }_3=\{(z_1+z_2+z_3,z_1{z}_2+z_2{z}_3+z_3{z}_1,z_1{z}_2{z}_3):|z_i|\le 1,i=1,2,3\}\subseteq {\mathbb{C}}^3$ $b{\Gamma }_3=\{(z_1+z_2+z_3,z_1{z}_2+z_2{z}_3+z_3{z}_1,z_1{z}_2{z}_3):|z_i|=1,i=1,2,3\}\subseteq {\Gamma }_3.$ A triple of commuting operators $(S_1,S_2,P)$ defined on a Hilbert space $\mathcal{H}$ for which ${\Gamma }_3$ is a spectral set is called a ${\Gamma }_3$-contraction. In this article we show by a counter example that there are ${\Gamma }_3$-contractions which do not dilate. It is also shown that under certain conditions a ${\Gamma }_3$-contraction can have normal $b{\Gamma }_3$ dilation. We determine several classes of ${\Gamma }_3$-contractions which dilate and show explicit construction of their dilations. A concrete functional model is provided for the ${\Gamma }_3$-contractions which dilate. Various characterizations for ${\Gamma }_3$-unitaries and ${\Gamma }_3$-isometries are obtained; the classes of ${\Gamma }_3$-unitaries and ${\Gamma }_3$-isometries are analogous to the unitaries and isometries in one variable operator theory. Also we find out a model for the class of pure ${\Gamma }_3$-isometries. En route we study the geometry of the sets ${\Gamma }_3$ and $b{\Gamma }_3$ and provide variety of characterizations for them.

Incomplete

References in corpus (1)

• Extremal holomorphic maps and the symmetrised bidisc

Cited by in corpus (1)

• A Schwarz lemma for the symmetrized tridisc and description of interpolating functions