Spectral analysis of certain groups of isometries on Hardy and Bergman spaces

Using the similarity theory of semigroups as well as spectral theory, we obtain the resolvents of the generators of strongly continuous groups of isometries on the Hardy and Bergman spaces. These groups are obtained as weighted composition operators associated with specific automorphisms of the upper half-plane. The resulting resolvents are given as integral operators for which we determine the norms and spectra.