Resumen de The double commutant property for composition operators

Miguel Lacruz Martín , Fernando León Saavedra , Srdjan Petrovic, Luis Rodríguez Piazza

• We investigate the double commutant property for a composition operator C_\varphi, induced on the Hardy space H^2({\mathbb {D}}) by a linear fractional self-map \varphi of the unit disk {\mathbb {D}}. Our main result is that this property always holds, except when \varphi is a hyperbolic automorphism or a parabolic automorphism. Further, we show that, in both of the exceptional cases, \{C_\varphi \}^{\prime \prime } is the closure of the algebra generated by C_\varphi and C_\varphi ^{-1}, either in the weak operator topology, if \varphi is a hyperbolic automorphism, or surprisingly, in the uniform operator topology, if \varphi 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.