Asymptotic Behavior of Polynomially Bounded Operators

Abstract

Let T be a polynomially bounded operator on a complex Banach space and let AT be the smallest uniformly closed (Banach) algebra that contains T and the identity operator. It is shown that for every S is an element of A(T), (n ->infinity)lim parallel to T(n)s parallel to = xi is an element of sigma(u)(T)sup vertical bar(S) over cap(xi)vertical bar where (S) over cap is the Gelfand transform of S and sigma(u) (T) := sigma(T) boolean AND Gamma is the unitary spectrum of T; Gamma = (z is an element of C: vertical bar z vertical bar = 1). (C) 2010 Academie des sciences. Published by Elsevier Masson SAS. All rights reserved.