Bernstein-Markov measures and Toeplitz theory

TL;DR

This paper establishes that Toeplitz operators linked to Bernstein-Markov measures on complex manifolds form an algebra and derives spectral equidistribution results, using asymptotic analysis of the Bergman kernel.

Contribution

It proves the algebra property of Toeplitz operators with Bernstein-Markov measures and applies this to spectral distribution analysis on complex manifolds.

Findings

Toeplitz operators form an algebra under composition

Spectral equidistribution results for these operators

Asymptotic analysis of the Bergman kernel

Abstract

We prove that Toeplitz operators associated with a Bernstein-Markov measure on a compact complex manifold endowed with a big line bundle form an algebra under composition. As an application, we derive a Szeg\H{o}-type spectral equidistribution result for this class of operators. A key component of our approach is the off-diagonal asymptotic analysis of the Bergman kernel, also known as the Christoffel-Darboux kernel.