Semigroups of composition operators on [formula omitted] spaces

Abstract

In this paper we prove that no non-trivial semigroup (φt)t≥0 consisting of analytic self-maps of the unit disk D generates a strongly continuous composition semigroup on Qp for p>0; that is, [φt,Qp]⫋Qp, which answers a question asked by A.G. Siskakis in 1996. We also give an affirmative answer to question in [9] that Qp,0⫋[φt,Qp] for all p>0 if the Denjoy-Wolff point of (φt)t≥0 is on the boundary of D, where Qp,0 is the closure of all polynomials in Qp space. Moreover, for a semigroup (φt)t≥0 with the Denjoy-Wolff point in D, we give a sufficient and necessary condition such that Qp,0⫋[φt,Qp] in terms of the infinitesimal generator of (φt)t≥0 for all p>0. As an application, a new characterization of Qp,0 in terms of semigroup (φt)t≥0 acting on Qp is given, which gives a generalization of the theorem of D. Sarason for VMOA.