A fredholm alternative-like result on power bounded operators

Abstract

Let X be a complex Banach space and T : X -> X be a power bounded operator, i.e., sup(n >= 0) parallel to T-m parallel to < infinity. We write B(X) for the Banach algebra of all bounded linear operators on X. We prove that the space Ran(I - T) is closed if and only if there exist a projection theta is an element of B(X) and an invertible operator R is an element of B(X) such that I - T = theta R = R theta. This paper also contains some consequences of this result.