A Fredholm alternative-like result on power bounded operators

Authors: ALİ ÜLGER, ONUR YAVUZ

Abstract: Let X be a complex Banach space and T:X\rightarrow 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 \theta\in B(X) and an invertible operator R \in B(X) such that I-T=\theta R=R\theta. This paper also contains some consequences of this result.

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