Date & Time: Thursday, October 30, 2014, 14:30-15:30.
Venue: Room 216
Title: Spectral sets and distinguished varieties in the symmetrized bidisc
Speaker: Sourav Pal, ISI Delhi
Abstract: This is a joint work with Orr Shalit. In this lecture, we shall try to explain a connection between complex geometry of the domain, the symmetrized bidisc and the pair of commuting matrices having the symmetrized bidisc as a spectral set. We show that for every pair of matrices $ (S,P) $, having the closed symmetrized bidisc $ \Gamma $ as a spectral set, there is a one dimensional complex algebraic variety $ \Lambda $ in $ \Gamma $ such that for every matrix valued polynomial $ f(z_1,z_2) $, $$ \|f(S,P)\|\leq \max_{(z_1,z_2) \in \Lambda}\|f(z_1,z_2)\|.$$
The variety $ \Lambda $ is shown to have the determinantal representation $$ \Lambda = \{(s,p) \in \Gamma : \det(F + pF^* - sI) = 0\} ,$$ where $F$ is the unique matrix of numerical radius not greater than 1 that satisfies $$ S-S^*P=(I-P^*P)^{\frac{1}{2}}F(I-P^*P)^{\frac{1}{2}}.$$
When $(S,P)$ is a strict $\Gamma$-contraction, then $\Lambda$ is a {\em distinguished variety} in the symmetrized bidisc, i.e. a one dimensional algebraic variety that exits the symmetrized bidisc through its distinguished boundary. We characterize all distinguished varieties of the symmetrized bidisc by a determinantal representation as above.