Higher Laplace-Beltrami Operators on Bounded Symmetric Domains

It was conjectured by the first author and Peetre that the higher Laplace.Beltrami operators generate the whole ring of invariant operators on bounded symmetric domains. We give a proof of the conjecture for domains of rank . 6 by using a graph manipulation of Kähler curvature tensor. We also compute higher order terms in the asymptotic expansions of the Bergman kernels and the Berezin transform on bounded symmetric domain.