A Szegö limit theorem for a class of Toeplitz operators on the Bergman space of the unit ball with singular symbols

Abstract

We obtain a Szegö limit theorem for a family of Toeplitz operators defined on the weighted Bergman space of the unit ball ${\mathbb{B}}_n$. The symbols of these operators are supported on some isotropic or co-isotropic submanifold $\Gamma \subseteq {\mathbb{B}}_n$ and can be seen, in general, as measures that are singular with respect to the Lebesgue measure on ${\mathbb{C}}^n$. The theorem given allows to describe the asymptotic behavior of these operators as the parameter of the weighted Bergman space tends to infinity.