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Spectrum of interpolated operators

Authors: Ernst Albrecht and Vladimir Müller
Journal: Proc. Amer. Math. Soc. 129 (2001), 807-814
MSC (2000): Primary 46B70, 47A10
Published electronically: September 20, 2000
MathSciNet review: 1804050
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Let $(X_0,X_1)$ be a compatible pair of Banach spaces and let $T$ be an operator that acts boundedly on both $X_0$ and $X_1$. 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.

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Additional Information

Ernst Albrecht
Affiliation: Fachbereich Mathematik, Universität des Saarlandes, Postfach 15 11 50, D–66041 Saarbrücken, Germany

Vladimir Müller
Affiliation: Institut of Mathematics AV ČR, Zitna 25, 115 67 Prague 1, Czech Republic

Keywords: Spectrum of interpolated operators, uniqueness-of-resolvent property
Received by editor(s): September 25, 1998
Received by editor(s) in revised form: May 14, 1999
Published electronically: September 20, 2000
Additional Notes: The second author was supported by the Alexander von Humboldt Foundation and partially by grant no. 201/96/0411 of GA ČR
Communicated by: David R. Larson
Article copyright: © Copyright 2000 American Mathematical Society