The Banach Algebra Generated by a Contraction

Abstract

Let T be a contraction on a Banach space and AT the Banach algebra generated by T. Let sigma(u)(T) be the unitary spectrum ( i. e., the intersection of sigma(T) with the unit circle) of T. We prove the following theorem of Katznelson-Tzafriri type: If sigma(u)(T) is at most countable, then the Gelfand transform of R is an element of A(T) vanishes on sigma(u)(T) if and only if lim(n ->infinity) parallel to(TR)-R-n parallel to = 0.