Absolutely Summing Toeplitz operators on Bergman spaces in the unit ball of ${\mathbb{C}}^n$

Abstract

In this paper, for $p>1$ and $r\ge 1$ we provide a complete characterization of the positive Borel measures $\mu$ on the unit ball $\B_n$ of ${\mathbb{C}}^n$ for which the induced Toeplitz operator $T_{\mu }$ is $r$-summing on the Bergman space $A^p$. We prove that the $r$-summing norm of $T_{\mu }:A^p\to A^p$ is equivalent to $\Vert \tilde{\mu }{\Vert }_{L^{\kappa }(d\lambda )}$, where $\kappa$ is a positive number determined by $p$ and $r$. As some preliminary, we describe when a Carleson embedding $J_{\mu }:A^p\to L^q(\mu )(1\le p,q\le 2)$ is $r$-summing, which extends the main result in [B. He, et al, Absolutely summing Carleson embeddings on Bergman spaces, Adv. Math., 439, 109495 (2024)].