Eigenvectors of Perturbed Operators

In this article, the perturbation of a compact, self-adjoint operator A, acting on a Hilbert space H is considered. This operator is perturbed by a family of self-adjoint operators of rank 1. Under some moderate restrictions, the existence of eigenvectors corresponding to the maximal eigenvalues of the perturbed operators is shown. These eigenvectors are subject to certain fixed normalization strategy and it is demanded that their norm is strictly decreasing