Commutants of the Pommiez operator

The Pommiez operator (∆ f )(z) = ( f (z)− f (0))/z is considered in the space (G) of the holomorphic functions in an arbitrary finite Runge domain G. A new proof of a representation formula of Linchuk of the commutant of ∆ in (G) is given. The main result is a representation formula of the commutant of the Pommiez operator in an arbitrary invariant hyperplane of (G). It uses an explicit convolution product for an arbitrary right inverse operator of ∆ or of a perturbation ∆− λI of it. A relation between these two types of commutants is found.