Composition operators whose symbols have orthogonal powers

Composition operators on the Hilbert Hardy space H² whose symbols are analytic selfmaps of the open unit disk having orthogonal powers are considered. The spectra and essential spectra of such operators are described. In the general case of an arbitrary analytic selfmap of the open unit disk, it is proved that the composition operator induced by that map has essential spectral radius less than 1 if and only if the map under consideration is a non-inner map with a fixed point in the unit disk. The canonical decomposition of a non-unitary composition contraction is determined.