Operator and function theory of the symmetrized polydisc
We establish necessary conditions, in the form of the positivity of Pick-matrices, for the existence of a solution to the spectral Nevanlinna-Pick problem. We approach this problem from an operator theoretic perspective. We restate the problem as an interpolation problem on the symmetrized polydisc Γ(κ). We establish necessary conditions for a κ-tuple of commuting operators to have Γ(κ) as a complete spectral set. We then derive necessary conditions for the existence of a solution of the spectral Nevanlinna- Pick problem. The final chapter of this thesis gives an application of our results to complex geometry. We establish an upper bound for the Caratheodory distance on int Γ(κ).