Unitary Equivalence of Composition C*-Algebras on the Hardy and Weighted Bergman Spaces

Let be an arbitrary linear-fractional self-map of the unit disk and consider the composition operator and the Toeplitz operator on the Hardy space and the corresponding operators and on the weighted Bergman spaces for . We prove that the unital C-algebra generated by and is unitarily equivalent to which extends a known result for automorphism-induced composition operators. For maps that are not automorphisms of , we show that is unitarily equivalent to , where and denote the ideals of compact operators on and , respectively, and apply existing structure theorems for to describe the structure of , up to isomorphism. We also establish a unitary equivalence between related weighted composition operators induced by maps that fix a point on the unit circle.