Spectral sets and distinguished varieties in the symmetrized bidisc

By Sourav Pal, Orr Shalit. 2013

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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, the norm of f(S,P) is less then the sup norm of f on $\Lambda$. The variety $\Lambda$ is shown to have a particular determinantal representation, related to the so-called "fundamental operator" of the pair (S,P). When (S,P) is a strict $\Gamma$-contraction, then $\Lambda$ is a 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.

Subfields: Algebra and Number Theory, Applied Mathematics, Mathematical Physics

View Paper: https://www.semanticscholar.org/paper/777ba6ad1fee9ee206cc9c540606c54e3851a060