The Bergman-space characterization of absolutely summing Toeplitz operators

Let

$$be a finite positive Borel measure on the unit ball$$

of $^n$, and let $1<p\backslashle 2$. Let $A^p$ denote the Bergman space, let T_ be the Toeplitz operator induced by

$$, let$$

be its Berezin transform, and let $K(,)$ be the Bergman kernel on $A^2$. The Bergman-space Toeplitz summability conjecture. The Toeplitz operator T_ is $r$-summing on $A^p$ if and only if

$$\int _{\mathbb B} \left|\widetilde{\mu}(z)^2\right|K(z, z) dv(z)<\infty.$$

This conjecture proposes a measure-theoretic characterization of absolutely summing Toeplitz operators on Bergman spaces, motivated by the analogous results for positive Toeplitz operators on Fock spaces. The general Bergman-space problem is presented as a direction for future investigation, and no resolution is supplied here.