Isometries of weighted Bergman-Privalov spaces on the unit ball of Cn

Abstract

Let $B$ denote the unit ball in $C^n$, and $v$ the normalized Lebesgue measure on $B$. For $\alpha >-1$, define $d{v}_{\alpha }\left(z\right)=\Gamma (n+\alpha +1)/\left\{\Gamma \right(n+1\left)\Gamma \right(\alpha +1\left)\right\}(1-|z{|}^2{)}^{\alpha }dv\left(z\right)$, $z\in B$. Let $H\left(B\right)$ denote the space of holomorphic functions in $B$. For $p\ge 1$, define

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${\left(\mathrm{AN}\right)}^p\left({\nu }_{\alpha }\right)$ is an $F$-space with respect to the metric $\rho \left(f,g\right)\equiv ∥f-g∥$. In this paper we prove that every linear isometry $T$ of ${\left(\mathrm{AN}\right)}^p\left({\nu }_{\alpha }\right)$ into itself is of the form $Tf=c\left(f\circ \psi \right)$ for all $f\in {\left(\mathrm{AN}\right)}^p\left({\nu }_{\alpha }\right)$ , where $c$ is a complex number with $\left|c\right|=1$ and $\psi$ is a holomorphic self-map of $B$ which is measure-preserving with respect to the measure ${\nu }_{\alpha }$.