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

TL;DR

This paper characterizes when Toeplitz operators on Bergman spaces in the unit ball of complex n-space are r-summing, linking their norms to measures and extending previous results on Carleson embeddings.

Contribution

It provides a complete characterization of r-summing Toeplitz operators on Bergman spaces, relating their norms to measure conditions and extending prior work on Carleson embeddings.

Findings

r-summing norm is equivalent to a specific L^κ norm of the measure

Characterization of measures for which Toeplitz operators are r-summing

Extension of results on Carleson embeddings to broader settings

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)].