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Bulletin of the London Mathematical Society 2001 33(1):77-82; doi:10.1112/blms/33.1.77
© 2001 by London Mathematical Society
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© London Mathematical Society

Spectral Characterization of Algebraic Elements

Thomas Ransford and Michael White

Département de mathématiques et de statistique, Université Laval Québec (QC), Canada G1K 7P4; e-mail: ransford{at}mat.ulaval.ca
Department of Mathematics, University of Newcastle Newcastle upon Tyne NE1 7RU; e-mail: michael.white{at}newcastle.ac.uk

Received 18 June 1999. Revision received 10 February 2000.

It is known that if a is an algebraic element of a Banach algebra A, then its spectrum {sigma}(a) is finite, and there exists {gamma} > 0 such that the Hausdorff distance to spectra of nearby elements satisfies

Formula
We prove that the converse is also true, provided that A is semisimple.

Research of first author supported by grants from NSERC (Canada) and the Fonds FCAR (Québec).


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