A Fredholm alternative-like result on power bounded operators

Let X be a complex Banach space and T:Xrightarrow X be a power bounded operator, i.e., sup_{n geq 0}|T^n| infty. We write B(X) for the Banach algebra of all bounded linear operators on X. We prove that the space range(I-T) is closed if and only if there exist a projection thetain B(X) and an invertible operator R in B(X) such that I-T=theta R=Rtheta. This paper also contains some consequences of this result.