Spectra of Composition Groups on the Weighted Dirichlet Space of the Upper Half-Plane

Abstract

We prove that the group of weighted composition operators induced by contin uous automorphism groups of the upper half plane U is strongly continuous on the weighted Dirichlet space of U, Dα(U). Further, we investigate when they are isometries on Dα(U). In each case, we determine the semigroup properties while in the case that the induced compo sition group is an isometry, we apply similarity theory to determine the spectral properties of the group.