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(z1,z2),‖f(S,P)‖⩽max(z1,z2)∈Λ‖f(z1,z2)‖. The variety Λ is shown to have the determinantal representationΛ={(s,p)∈Γ:det(F+pF⁎−sI)=0}, where F is the unique matrix of numerical radius not greater than 1 that satisfiesS−S⁎P=(I−P⁎P)12F(I−P⁎P)12. 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.