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 s (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 l2 for solving some PDEs, extended to general operator space under some conditions here.