The spectral Caratheodory-Fejer problem

Huang-Nan Huang, Stefania Marcantognini and Nicholas Young

Keywords

spectral radius, symmetrised polydisc, Kobayashi metric

Status

To appear in Integral Equations and Operator Theory.

Abstract

The problem is: given k by k matrices V_0,...,V_n, determine whether there exists an analytic k by k matrix-valued function F in the unit disc such that the jth derivative of F at 0 is V_j for j=0,1,...,n and F(z) has spectral radius less than 1 for every z in the disc. We show that the problem can be reduced to an analogous interpolation problem for analytic functions from the disc to the symmetrised polydisc G_k. We show that in the case n=1 the resulting problem is equivalent to the infinitesimal Kobayashi extremal problem for G_k. We give a concrete solution in the case k=2, n=1. We also obtain a different type of necessary and sufficient condition for the existence of F in the case k=2, and we present a necessary condition for the existence of F in the case of general k and n.

The paper is available in pdf format here

Nicholas Young
6 June 2005.