Commutants of the Pommiez operator

Abstract

The Pommiez operator(Δf)(z)=(f(z)−f(0))/zis considered in the spaceℋ(G)of the holomorphic functions in an arbitrary finite Runge domainG. A new proof of a representation formula of Linchuk of the commutant ofΔinℋ(G)is given. The main result is a representation formula of the commutant of the Pommiez operator in an arbitrary invariant hyperplane ofℋ(G). It uses an explicit convolution product for an arbitrary right inverse operator ofΔor of a perturbationΔ−λIof it. A relation between these two types of commutants is found.