On a theorem of Gelfand and its local generalizations
Tom 123 / 1997
Studia Mathematica 123 (1997), 185-194
DOI: 10.4064/sm-123-2-185-194
Streszczenie
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, $T_1,..., T_m$, on a Banach space X.
