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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. |