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Title

On a theorem of Gelfand and its local generalizations

Authors 1

Affiliations

  1. Department of Mathematics and Computer Science, Faculty of Science, Kuwait University, P.O. Box 5969 Safat 13060, Kuwait

Abstract

In 1941, I. Gelfand proved that if a is a doubly power-bounded element of a Banach algebra A such that Sp(a) = {1}, then a = 1. In [4], this result has been extended locally to a larger class of operators. In this note, we first give some quantitative local extensions of Gelfand-Hille's results. Secondly, using the Bernstein inequality for multivariable functions, we give short and elementary proofs of two extensions of Gelfand's theorem for m commuting bounded operators, T1,...,Tm, on a Banach space X.

Keywords

ENG
locally power-bounded operator, local spectrum, local spectral radius

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Studia Mathematica
1997/123/2
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Pages:
185-194
Main language of publication
English
Received
1996-02-20
Accepted
1996-09-24
Published
1997
Exact and natural sciences