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Local spectral properties of commutators

Published online by Cambridge University Press:  20 January 2009

Kjeld B. Laursen ,
Vivien G. Miller  and
Michael M. Neumann
Kjeld B. Laursen
Affiliation:
Mathematics Institute, University of Copenhagen, Universitetsparken 5, 2100 Copenhagen 0, Denmark
Vivien G. Miller
Affiliation:
Department of Mathematics and Statistics, Mississippi State University, Mississippi State, MS 39762, U.S.A.
Michael M. Neumann
Affiliation:
Department of Mathematics and Statistics, Mississippi State University, Mississippi State, MS 39762, U.S.A.
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Abstract

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For a pair of continuous linear operators T and S on complex Banach spaces X and Y, respectively, this paper studies the local spectral properties of the commutator C(S, T) given by C(S, T)(A): = SA−AT for all A∈L(X, Y). Under suitable conditions on T and S, the main results provide the single valued extension property, a description of the local spectrum, and a characterization of the spectral subspaces of C(S, T), which encompasses the closedness of these subspaces. The strongest results are obtained for quotients and restrictions of decomposable operators. The theory is based on the recent characterization of such operators by Albrecht and Eschmeier and extends the classical results for decomposable operators due to Colojoară, Foiaş, and Vasilescu to considerably larger classes of operators. Counterexamples from the theory of semishifts are included to illustrate that the assumptions are appropriate. Finally, it is shown that the commutator of two super-decomposable operators is decomposable.

Type
Research Article
Information
Proceedings of the Edinburgh Mathematical Society , Volume 38 , Issue 2 , June 1995 , pp. 313 - 329
Copyright
Copyright © Edinburgh Mathematical Society 1995

References

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