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 matrix or a linear operator under finite rank perturbations is proved: The increase of dimension from the $n$-th power of the kernel of the perturbed operator to the $(n+1)$-th power differs from the increase of dimension of the corresponding powers of the kernels of the unperturbed operator by at most the rank of the perturbation and this bound is sharp. 
Jussi Behrndt, Leslie Leben, Francisco Mart\'inez Per\'ia, and Carsten   Trunk
math.FA