ON A THEOREM OF GELFAND AND ITS LOCAL GENERALIZATIONS

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 t his note, we first give some quantitative local extensions of Gelfand- Hille's results. Secondly, using the Bernstein inequality for multivar iable 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.