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, , on a Banach space X.
@article{bwmeta1.element.bwnjournal-article-smv123i2p185bwm,
author = {Driss Drissi},
title = {On a theorem of Gelfand and its local generalizations},
journal = {Studia Mathematica},
volume = {122},
year = {1997},
pages = {185-194},
zbl = {0894.47018},
language = {en},
url = {http://dml.mathdoc.fr/item/bwmeta1.element.bwnjournal-article-smv123i2p185bwm}
}
Drissi, Driss. On a theorem of Gelfand and its local generalizations. Studia Mathematica, Tome 122 (1997) pp. 185-194. http://gdmltest.u-ga.fr/item/bwmeta1.element.bwnjournal-article-smv123i2p185bwm/
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