Discrete Dynamics in Nature and Society
Volume 2009 (2009), Article ID 725860, 16 pages
Research Article

Isometries of a Bergman-Privalov-Type Space on the Unit Ball

1Mathematical Institute of the Serbian Academy of Sciences, Knez Mihailova 36/III, 11000 Beograd, Serbia
2Faculty of Engineering, Ibaraki University, Hitachi 316-8511, Japan

Received 14 July 2009; Accepted 9 September 2009

Academic Editor: Leonid Berezansky

Copyright © 2009 Stevo Stević and Sei-Ichiro Ueki. 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.


We introduce a new space ANlog,α(𝔹) consisting of all holomorphic functions on the unit ball 𝔹n such that fANlog,α:=𝔹φe(ln(1+|f(z)|))dVα(z)<, where α>1, dVα(z)=cα,n(1|z|2)αdV(z) (dV(z) is the normalized Lebesgue volume measure on 𝔹, and cα,n is a normalization constant, that is, Vα(𝔹)=1), and φe(t)=tln(e+t) for t[0,). Some basic properties of this space are presented. Among other results we proved that ANlog,α(𝔹) with the metric d(f,g)=fgANlog,α is an F-algebra with respect to pointwise addition and multiplication. We also prove that every linear isometry T of ANlog,α(𝔹) into itself has the form Tf=c(fψ) for some c such that |c|=1 and some ψ which is a holomorphic self-map of 𝔹 satisfying a measure-preserving property with respect to the measure dVα. As a consequence of this result we obtain a complete characterization of all linear bijective isometries of ANlog,α(𝔹).