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This is our third work on Bergman-type operator over bounded domains. In theprevious two articles, we systematically study the boundedness, compactness andSchatten membership of Bergman-type on the Hilbert unit ball. In the presentpaper, we investigate singular integral operators induced by the Bergman kerneland Szeg\"o kernel on the irreducible bounded symmetric domain in its standardHarish-Chandra realization. We completely characterize when Bergman-typeoperators and Szeg\"o-type operators belong to Schatten class operator idealsby several analytic numerical invariants of the bounded symmetric domain. Theseresults generalize a recent result on the Hilbert unit ball due to the authorand his coauthor but also cover all irreducible bounded symmetric domains.Moreover, we obtain two trace formulae and a new integral estimate related tothe Forelli-Rudin estimate. The key ingredient of the proofs involves thefunction theory on the bounded symmetric domain and the spectrum estimate ofBergman-type and and Szeg\"o-type operators.

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