MEROMORPHIC RESOLVENTS

by Olavi Nevanlinna Olavi.Nevanlinna@hut.fi Helsinki University of Technology P.O.Box 1100 FIN-02015 HUT Finland


Abstract

Scalar meromorphic functions can have Picard special values, that is, values which are not taken at all. More generally, certain values can be defective but the sum of all defects is bounded. We have studied operator valued meromorphic functions, developed a perturbation theory for them (Ann. Acad. Sci.Fenn. Mathematica 2000, Vol 25, 3-30), which can be used to show for example that growth of resolvents as meromorphic functions is robust under small rank perturbations. We have published earlier that the growth of resolvent can be linked to speed of iterations (together with Saara Hyv"onen, BIT, 2000, vol 40, 267-290). In this talk we link the Picard defect values to "nonnormality", by showing that if the operator is normal and the growth of the resolvent is of finite order, then the resolvent is not defective in the sense of value distribution. We also give a result of the following general nature. Suppose f(z) is a scalar meromorphic function and A is a bounded operator with a meromorphic resolvent, then f(zA) is an operator valued meromorphic function and its growth is is on the level of the maximum of the that of f and the resolvent. The new results are intended to appear in a monograph in the AMS Fields Institute monograph series.