Composition operators with a minimal commutant

Let Cφ be a composition operator on the Hardy space H2, induced by a linear fractional self-map φ of the unit disk. We consider the question whether the commutant of Cφ is minimal, in the sense that it reduces to the weak closure of the unital algebra generated by Cφ. 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 Königs's domain is bounded and strictly starlike with respect to the origin, and we show that the operator Cφ has a minimal commutant. Furthermore, we provide two examples of univalent, analytic self-maps φ of the unit disk such that Cφ is compact but it fails to have a minimal commutant.