Schatten classes of Volterra operators on Bergman-type spaces in the unit ball

Abstract

<p style='text-indent:20px;'>We devote to studying the condition of a holomorphic function <inline-formula><tex-math id="M1">\begin{document}$ g $\end{document}</tex-math></inline-formula> in the complex unit ball <inline-formula><tex-math id="M2">\begin{document}$ \mathbb{B}_n $\end{document}</tex-math></inline-formula> so that the Volterra operator <inline-formula><tex-math id="M3">\begin{document}$ T_g:A_\alpha^2\to A_\alpha^2 $\end{document}</tex-math></inline-formula> belongs to the Schatten <inline-formula><tex-math id="M4">\begin{document}$ p $\end{document}</tex-math></inline-formula>-class. Assuming <inline-formula><tex-math id="M5">\begin{document}$ n\ge2 $\end{document}</tex-math></inline-formula> and <inline-formula><tex-math id="M6">\begin{document}$ \alpha&gt;-3 $\end{document}</tex-math></inline-formula>, we get the following conclusions</p><p style='text-indent:20px;'>1. For <inline-formula><tex-math id="M7">\begin{document}$ 0&lt;p\le n $\end{document}</tex-math></inline-formula>, <inline-formula><tex-math id="M8">\begin{document}$ T_g\in \mathcal{S}_p(A^2_\alpha) $\end{document}</tex-math></inline-formula> if and only if <inline-formula><tex-math id="M9">\begin{document}$ g $\end{document}</tex-math></inline-formula> is a constant. </p><p style='text-indent:20px;'>2. For <inline-formula><tex-math id="M10">\begin{document}$ n&lt;p&lt;\infty $\end{document}</tex-math></inline-formula> and <inline-formula><tex-math id="M11">\begin{document}$ p(\alpha+1)+4n&gt;0 $\end{document}</tex-math></inline-formula>, <inline-formula><tex-math id="M12">\begin{document}$ T_g\in \mathcal{S}_p(A^2_\alpha) $\end{document}</tex-math></inline-formula> if and only if</p><p style='text-indent:20px;'><disp-formula> <label/> <tex-math id="FE1"> \begin{document}$ \begin{align*} \int_{\mathbb{B}_n}\left((1-|w|^2)^{n+1+\alpha+2t} \int_{\mathbb{B}_n} \frac{|Rg(z)|^2 \mathrm{d} v_{\alpha+2}(z)}{|1-\langle z, w\rangle|^{2(n+1+\alpha+t)}}\right)^\frac p2 { \mathrm{d} \tau(w)} &lt;\infty, \end{align*} $\end{document} </tex-math></disp-formula></p><p style='text-indent:20px;'>where <inline-formula><tex-math id="M13">\begin{document}$ t&gt;\max\{\frac np-\frac{n+1+\alpha}2, \frac{n-1}2\} $\end{document}</tex-math></inline-formula> and and <inline-formula><tex-math id="M14">\begin{document}$ \mathrm{d} \tau(w) = (1-|w|^2)^{-n-1}{ \mathrm{d} v(w)} $\end{document}</tex-math></inline-formula> is the Möbius invariant measure in <inline-formula><tex-math id="M15">\begin{document}$ \mathbb{B}_n $\end{document}</tex-math></inline-formula>. Here <inline-formula><tex-math id="M16">\begin{document}$ \mathrm{d} v $\end{document}</tex-math></inline-formula> is the normalized Lebesgue measure on <inline-formula><tex-math id="M17">\begin{document}$ \mathbb{B}_n $\end{document}</tex-math></inline-formula> so that <inline-formula><tex-math id="M18">\begin{document}$ v( \mathbb{B}_n) = 1 $\end{document}</tex-math></inline-formula> and <inline-formula><tex-math id="M19">\begin{document}$ \mathrm{d} v_{\alpha+2}(z) = c_{\alpha+2}(1-|z|^2)^{\alpha+2} \mathrm{d} v (z) $\end{document}</tex-math></inline-formula> with a normalized constant <inline-formula><tex-math id="M20">\begin{document}$ c_{\alpha+2} $\end{document}</tex-math></inline-formula> so that <inline-formula><tex-math id="M21">\begin{document}$ v_{\alpha+2}( \mathbb{B}_n) = 1 $\end{document}</tex-math></inline-formula>.</p>