Semigroups of analytic functions and composition operators.

Abstract

The authors consider homomorphisms t→φt of R+ (considered additively) into the semigroup of holomorphic maps of a given open set U(⊂C) into itself which have the property that φ:(z,t)↦φt(z) is continuous on U×R+. It is shown that there exists a unique holomorphic function A on U such that ∂φ/∂t=A∘φ. The functions A are characterized for the case U={Rez>0}. The strongly continuous one-parameter semigroups of composition operators on Hp(Δ), 1≤p