The double commutant property for composition operators

Abstract

We investigate the double commutant property for a composition operator ?? , induced on the Hardy space ?2(?) by a linear fractional self-map ? of the unit disk ?. Our main result is that this property always holds, except when ? is a hyperbolic automorphism or a parabolic automorphism. Further, we show that, in both of the exceptional cases, {??}′′ is the closure of the algebra generated by ?? and ?−1?, either in the weak operator topology, if ? is a hyperbolic automorphism, or surprisingly, in the uniform operator topology, if ? is a parabolic automorphism. Finally, for each type of a linear fractional mapping, we settle the question when any of the algebras involved are equal.