Operator and functional theory of the symmetrized polydisc

Abstract

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: Let k and n be natural numbers. Choose n distinct points zj in the open unit disc, D, and n matrices Wj in Mk(C), the space of complex k × k matrices. Does there exist an analytic function : D ! Mk(C) such that (zj ) = Wj for j = 1, ...., n and ( (z)) D for all z 2 D? We approach this problem from an operator theoretic perspective. We restate the problem as an interpolation problem on the symmetrized polydisc 􀀀k, 􀀀k = {(c1(z), . . . , ck(z)) | z 2 D} Ck where cj(z) is the jth elementary symmetric polynomial in the components of z. We establish necessary conditions for a k-tuple of commuting operators to have 􀀀k 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 􀀀k.