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 $\|\widetilde{\mu}\|_{L^{\kappa}(d\lambda)}$, 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)].