Duality of the Nonreflexive Bergman Space of the Upper Half Plane and Composition Groups

Abstract

We identify the predual of the nonreflexive Bergman space of the upper half plane with the little Bloch space of the upper half plane consisting of those functions vanishing at point i. Using the duality pairing as well as the composition groups on the nonreflexive Bergman space, we obtain the groups of composition operators defined on the identified predual. We identify the infinitesimal generator of each group and prove the strong continuity property. We then obtain the spectra of the generator Γ, determine the resolvents and further obtain the spectra and the norms of the resulting resolvents.