Article ID | Journal | Published Year | Pages | File Type |
---|---|---|---|---|
6415192 | Journal of Functional Analysis | 2014 | 22 Pages |
We show that for every pair of matrices (S,P), having the closed symmetrized bidisc Î as a spectral set, there is a one dimensional complex algebraic variety Î in Î such that for every matrix valued polynomial f(z1,z2),âf(S,P)â⩽max(z1,z2)âÎâf(z1,z2)â. The variety Î is shown to have the determinantal representationÎ={(s,p)âÎ:det(F+pFââsI)=0}, where F is the unique matrix of numerical radius not greater than 1 that satisfiesSâSâP=(IâPâP)12F(IâPâP)12. When (S,P) is a strict Î-contraction, then Î 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.