Bergman kernel estimates and Toeplitz operators on holomorphic line bundles

TL;DR

This paper characterizes key properties of Toeplitz operators on Bergman spaces of holomorphic line bundles over Kähler Cartan-Hadamard manifolds, linking operator behavior to geometric and measure-theoretic properties.

Contribution

It provides a new characterization of Toeplitz operator properties in terms of geometric and measure-theoretic conditions on Kähler manifolds.

Findings

Operator properties are characterized by measure conditions.

Boundedness, compactness, and Schatten class membership are linked to geometric measures.

Results extend to Bergman spaces on complex manifolds.

Abstract

We characterize operator-theoretic properties (boundedness, compactness, and Schatten class membership) of Toeplitz operators with positive measure symbols on Bergman spaces of holomorphic hermitian line bundles over K\"ahler Cartan-Hadamard manifolds in terms of geometric or operator-theoretic properties of measures.