ON THE RELATIONS BETWEEN THE POINT SPECTRUM OF A AND INVERTIBILITY OF I + f(A)B

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