On the relations between the point spectrum of A and invertibility of I + f(A)B

Abstract

Let A be a bounded linear operator on a Banach space X. We <br />investigate the conditions of existing rank-one operator B such that I+f(A)B is invertible for every analytic function <br />f on sigma(A). Also we compare the invariant subspaces of f(A)B and B. This work is motivated by an <br />operator method on the Banach space ell^2 for solving some PDEs which is extended to general operator space under some <br />conditions in this paper.