Approximate solutions of problems involving normal operators (Q1097485)

The paper deals with approximate solutions of problems with an operator realization of the form \(Tu=f\in H\) where H is a Hilbert space and T is an unbounded normal operator. In case T has the form \(\lambda\) I-A with A a bounded operator of sufficiently small norm, then usual approximation procedures involve a polynomial of powers of A. When A is unbounded then the spectrum of A becomes unbounded. However, if the spectral theorem can be applied to A, then under certain restrictions the spectrum of \(T^{- 1}\) and \(A^{-1}\) both become bounded. This fact is used as a basis for devising an approximation procedure based on a polynomial of powers of \(A^{-1}\). The method presented in this paper can be used for approximation of more general operators than its inverse in terms of more easily computable functions, leading to a priori error estimates. Particular results are derived for resolvents and semigroups in terms of resolvents.