Local Spectral Theory for Operators on BanachSpaces
نویسنده
چکیده
In this article, we shall discuss some recent developments and applications of the local spectral theory for linear operators on Banach spaces. Special emphasis will be given to those parts of operator theory, where spectral theory, harmonic analysis, and the theory of Banach algebras overlap and interact. Along this line, we shall present the recent progress of the theory of quotients and restrictions of decomposable operators, some connections between local spectral theory and the Kato resolvent set, and a general theory of spectral inclusions for certain parts of the spectrum. The abstract theory will be applied and exempliied in the context of convolution operators induced by measures on a locally compact abelian group G: In particular, we shall investigate those measures on G; for which the corresponding convolution operators on the group algebra L 1 (G) or the measure algebra M (G) are decomposable in the sense of Foia s or have a natural spectrum in the sense of Zafran.
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