LAUSR

Abstract Let C φ be a composition operator on the Hardy space H 2 , induced by a linear fractional self-map φ of the unit disk. We consider the question whether the commutant of C φ is minimal, in the sense that it reduces to the weak closure of the unital algebra generated by C φ . We show that this happens in exactly three cases: when φ is either a non-periodic elliptic automorphism, or a parabolic non-automorphism, or a loxodromic self-map of the unit disk. Also, we consider the case of a composition operator induced by a univalent, analytic self-map φ of the unit disk that fixes the origin and that is not necessarily a linear fractional map, but in exchange its Konigs's domain is bounded and strictly starlike with respect to the origin, and we show that the operator C φ has a minimal commutant. Furthermore, we provide two examples of univalent, analytic self-maps φ of the unit disk such that C φ is compact but it fails to have a minimal commutant.