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

Abstract

We introduce a new spaceANlog⁡,α(&#x1D539;)consisting of all holomorphic functions on the unit ball&#x1D539;⊂ℂnsuch that‖f‖ANlog⁡,α:=∫&#x1D539;φe(ln⁡(1+|f(z)|))dVα(z)<∞, whereα>−1,dVα(z)=cα,n(1−|z|2)αdV(z)(dV(z)is the normalized Lebesgue volume measure on&#x1D539;, andcα,nis a normalization constant, that is,Vα(&#x1D539;)=1), andφe(t)=tln⁡(e+t)fort∈[0,∞). Some basic properties of this space are presented. Among other results we proved thatANlog⁡,α(&#x1D539;)with the metricd(f,g)=‖f−g‖ANlog⁡,αis anF-algebra with respect to pointwise addition and multiplication. We also prove that every linear isometryTofANlog⁡,α(&#x1D539;)into itself has the formTf=c(f∘ψ)for somec∈ℂsuch that|c|=1and someψwhich is a holomorphic self-map of&#x1D539;satisfying a measure-preserving property with respect to the measuredVα. As a consequence of this result we obtain a complete characterization of all linear bijective isometries ofANlog⁡,α(&#x1D539;).