International Journal of Mathematics and Mathematical Sciences
Volume 2005 (2005), Issue 8, Pages 1239-1251
doi:10.1155/IJMMS.2005.1239
Commutants of the Pommiez operator
Ivan H. Dimovski
and Valentin Z. Hristov
Department of Complex Analysis, Institute of Mathematics and Informatics, Bulgarian Academy of Sciences, Acad. G. Bonchev Street Block 8, Sofia 1113, Bulgaria
Abstract
The Pommiez operator (Δf)(z)=(f(z)−f(0))/z is considered in the space ℋ(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 Δ in ℋ(G) is given. The main result is a representation formula of the commutant of the Pommiez operator in an arbitrary invariant hyperplane of ℋ(G). It uses an explicit convolution product for an arbitrary right inverse operator of Δ or of a perturbation Δ−λI of it. A relation between these two types of commutants is found.