Proceedings of the Edinburgh Mathematical Society (Series 2)

Table of Contents - June 1995 - Volume 38, Issue 02  

Research Article

Local spectral properties of commutators

Kjeld B. Laursena1, Vivien G. Millera2 and Michael M. Neumanna2

a1 Mathematics Institute, University of Copenhagen, Universitetsparken 5, 2100 Copenhagen 0, Denmark

a2 Department of Mathematics and Statistics, Mississippi State University, Mississippi State, MS 39762, U.S.A.

Abstract

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 Axs2208L(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.

(Received November 01 1993)