Absolutely summing Hankel operators on Bergman spaces

Abstract

In this paper we initiate the study of absolute summability for big and little Hankel operators $H_f^2,h_f^2:A_{\hat{I}}{\pm }^p({\mathbb{B}}_n)\to L^q({\mathbb{B}}_n,d{v}_{\hat{I}}^2),$ acting between weighted Bergman and weighted Lebesgue spaces on the unit ball, for possibly different integrability exponents $p$ and $q$. We characterize those symbols $f$ for which the big Hankel operator $H_f^2$ is $r$-summing, and those for which the little Hankel operator $h_f^2$ is $r$-summing. Our approach relies on a deep revisit of the absolute summability of the associated Carleson embedding operators from $A_{\hat{I}}{\pm }^p({\mathbb{B}}_n)$ to $L^q({\mathbb{B}}_n,d{v}_{\hat{I}}^2)$, from which we obtain characterizations of absolutely summing big and little Hankel operators that appear to be new even in the diagonal case $p=q$.

34 pages