Spectral sets and distinguished varieties in the symmetrized bidisc

We show that for every pair of matrices (S,P), having the closed symmetrized bidisc $Γ$ as a spectral set, there is a one dimensional complex algebraic variety $Λ$ in $Γ$ such that for every matrix valued polynomial f, the norm of f(S,P) is less then the sup norm of f on $Λ$. The variety $Λ$ 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 $Γ$-contraction, then $Λ$ 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.

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