On domains of powers of linear operators and finite rank perturbations

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 domT n 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. Mathematics Subject Classification (2000). Primary 47A05; Secondary 47B50.