  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.
