Invariant operators and the Berezin transform on Cartan domains

Let Omega be an irreducible Cartan domain of rank r and genus p and B-nu (nu > p - 1) be the Berezin transform on Omega. It is known that as nu tends to infinity, the Berezin transform admits the asymptotic expansion B-nu approximate to Sigma(k=o)(infinity)Q(k)nu(-k) where the Q(k)'s are certain invariant differential operators - for instance, Q(0) is the identity and Q(1) is the Laplace-Beltrami operator. [See A. UNTERBERGER and H. UPMEIER, Comm. Math. Phys. 164 (1994), 563-598.] In the present paper we show that the operators Q(k) generate the whole ring of invariant differential operators on Omega; in fact, Q(1), Q(3)...,Q(2r-1) and Q(0) form a set of free generators. A bounded version of this result is also given: for any nu > p-1, the r+1 operators B-nu, Bnu+1,..., Bnu+r are a set of generators for the von Neumann algebra 3 of all G-invariant bounded linear operators on L-2(Omega); this algebra can be identified with the algebra of all L-2 - bounded Fourier multipliers, or of all bounded operators which are functions, in the L-2 - spectral - theoretic sense, of certain normal extensions of the invariant differential operators.