We determine the spectra of weighted composition operators acting on the weighted Banach spaces of analytic functions H∞ vp when the symbol φ has a fixed point in the open unit disk. Further, we apply this result to give the spectra of composition operators on Bloch type spaces. In particular, we answer in the affirmative a conjecture by MacCluer and Saxe.

Natural metrics in the hyperbolic α-Bloch-, weighted Dirichletand Qp-classes are introduced, and these classes are shown to be complete metric spaces with respect to the corresponding metrics. Then Lipschitz continuous and compact composition operators Cφ(f) = f ◦φ acting from the hyperbolic α-Bloch-class to the hyperbolic weighted Dirichletor Qp-class are characterized by conditions depending ...