Norm estimates for functions of non-selfadjoint operators nonregular on the convex hull of the spectrum

We consider a bounded linear operator A in a Hilbert space with a Hilbert-Schmidt Hermitian component (A − A*)/2i. A sharp norm estimate is established for functions of A nonregular on the convex hull of the spectrum. The logarithm, fractional powers and meromorphic functions of operators are examples of such functions. Our results are based on the existence of a sequence An (n = 1, 2, ...) of finite dimensional operators strongly converging to A, whose spectra belongs to the spectrum of A. Besides, it is shown that the resolvents and holomorphic functions of An strongly converge to the resolvent and corresponding function of A.