Commutants of certain composition operators

If [special characters omitted] is an analytic map from the unit disk into itself then the composition operator [special characters omitted] can be defined on the Hardy space by [special characters omitted]. We study the commutant of composition operators, that is, the set of bounded operators on the Hardy space that commute with the given composition operator. Non-trivial examples of maps are given such that the commutant of the induced composition operator strictly contains the weak operator closure of the algebra generated by the Toeplitz and composition operator. These examples encompass both the case when the map [special characters omitted] has an interior fixed point and the case when the Denjoy-Wolff point is on the boundary of the disk. The commutant of certain compact composition operators is characterized. This characterization allows us to conclude these commutants are commutative and enables us to begin comparing commutants of composition operators