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
Department of Complex Analysis, Institute of Mathematics and Informatics, Bulgarian Academy of Sciences, Acad. G. Bonchev Street Block 8, Sofia 1113, Bulgaria
Received 10 October 2004
Copyright © 2005 Ivan H. Dimovski and Valentin Z. Hristov. This is an open access article distributed under the Creative Commons Attribution License, which permits unrestricted use, distribution, and reproduction in any medium, provided the original work is properly cited.
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.