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.