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Spectrum of interpolated operators

Authors:	Ernst Albrecht Vladimir Mü ller

Institution:	Fachbereich Mathematik, Universität des Saarlandes, Postfach 15 11 50, D--66041 Saarbrücken, Germany ; Institut of Mathematics AV CR, Zitna 25, 115 67 Prague 1, Czech Republic

Abstract:	Let be a compatible pair of Banach spaces and let be an operator that acts boundedly on both and . Let $T_{\theta]} \quad(0\le\theta\le 1)$ be the corresponding operator on the complex interpolation space $(X_0,X_1)_{\theta]}$ . The aim of this paper is to study the spectral properties of $T_{\theta]}$ . We show that in general the set-valued function $\theta\mapsto \sigma(T_{\theta]})$ is discontinuous even in inner points $\theta\in(0,1)$ and show that each operator satisfies the local uniqueness-of-resolvent condition of Ransford. Further we study connections with the real interpolation method.

Keywords:	Spectrum of interpolated operators uniqueness-of-resolvent property

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