The effect of finite rank perturbations on Jordan chains of linear operators

A general result on the structure and dimension of the root subspaces of a linear operator under finite rank perturbations is proved: The increase of dimension from the kernel of the n-th power to the kernel of the (n+1)-th power of the perturbed operator differs from the increase of dimension of the kernels of the corresponding powers of the unperturbed operator by at most the rank of the perturbation. This bound is sharp.