Schatten class Bergman-type and Szeg\"o-type operators on bounded symmetric domains

TL;DR

This paper characterizes when Bergman-type and Szeg"o-type operators on irreducible bounded symmetric domains belong to Schatten classes, extending previous results from the unit ball to all such domains using spectral and function theory techniques.

Contribution

It provides a complete characterization of Schatten class membership for these operators on all irreducible bounded symmetric domains, generalizing prior work on the unit ball.

Findings

Complete Schatten class characterization for Bergman-type and Szeg"o-type operators.

Derivation of two trace formulae and a new integral estimate.

Extension of previous results from the unit ball to all irreducible bounded symmetric domains.

Abstract

This is our third work on Bergman-type operator over bounded domains. In the previous two articles, we systematically study the boundedness, compactness and Schatten membership of Bergman-type on the Hilbert unit ball. In the present paper, we investigate singular integral operators induced by the Bergman kernel and Szeg\"o kernel on the irreducible bounded symmetric domain in its standard Harish-Chandra realization. We completely characterize when Bergman-type operators and Szeg\"o-type operators belong to Schatten class operator ideals by several analytic numerical invariants of the bounded symmetric domain. These results generalize a recent result on the Hilbert unit ball due to the author and his coauthor but also cover all irreducible bounded symmetric domains. Moreover, we obtain two trace formulae and a new integral estimate related to the Forelli-Rudin estimate. The key ingredient of the proofs involves the function theory on the bounded symmetric domain and the spectrum estimate of Bergman-type and and Szeg\"o-type operators.