Abstract: We prove that the group of weighted composition operators induced by continuous automorphism groups of the upper half plane $\mathbb{U}$ is strongly continuous on the weighted Dirichlet space of $\mathbb{U}$, $\mathcal{D}_{α}$($\mathbb{U}$). Further, we investigate when they are isometries on $\mathcal{D}_{α}$($\mathbb{U}$). In each case, we determine the semigroup properties while in the case that the induced composition group is an isometry, we apply similarity theory to determine the spectral properties of the group.

Key words: one-parameter semigroup, composition operator semigroups, strong continuity, infinitesimal generator, spectra