Commutants of composition operators

grantor: University of TorontoLet $C\sb{\phi}$ be a composition operator on $H\sp2(D).$ We consider which operators A commute with $C\sb{\phi}.$ In particular, we prove that, unless $\phi$ is an elliptic disc automorphism of finite periodicity, the only Toeplitz operators that commute must be analytic. Also, we provide a classification of the multiplication operators that commute for several particular cases of $\phi .$ We prove that the outer functions are an invariant class for the composition operators. In the last two chapters we provide some results about the commutant in general when $\phi$ has an interior fixed point. In particular, we find some hyperinvariant subspaces of such operators. We also obtain a partial classification of the quasi-normal composition operators.Ph.D