Let S and T be linear operators in a linear space such that S T. In this note an estimate for the codimension of domSn in domTn in terms of the codimension of domS in domT is obtained. An immediate consequence is that for any polynomial p the operator p(S) is a finite-dimensional restriction of the operator p(T) whenever S is a finite-dimensional restriction of T. The general results are applied to a perturbation problem of selfadjoint definitizable operators in Krein spaces.