Spectral sets and distinguished varieties in the symmetrized bidisc.

Abstract: We show that for every pair of matrices , 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 , The variety Λ is shown to have the determinantal representation where F is the unique matrix of numerical radius not greater than 1 that satisfies When 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. [Copyright &y& Elsevier]