Let R be an irreducible Cartan domain of rank r and genus p and B, (Y > p -1) be the Berezin transform on R. It is known that as v tends to infinity, the Berezin transform admits the asymptotic expansion B, M C' p,, Q k U k where the Qk'S are certain invariant differential operatorsfor instance, Qo is the identity and Q1 is the Laplace-Beltrami operator. [See A. UN-TERBERGER and H. UPMEIER, Comm. Math. Phys. 164 (1994), 563-598.1 In the present paper we show that the operators Qk generate the whole ring of invariant differential operators on R; in fact, Q 1 , Q3,. . . , Q2?-1 and Qo form a set of free generators. A bounded version of this result is also given: for any v > p -1, the r + I operators B,, B Y + ] ,
. . . , B,+? are a set of generators for the von Neumann algebra 3 of all G-invariant bounded linear operators on L2(R); this algebra can be identified with the algebra of all L2 -bounded Fourier multipliers, or of all bounded operators which are functions, in the Lz -spectraltheoretic sense, of certain normal extensions of the invariant differential operators.
1. Invariant differential operators
Let R = G / K be an irreducible Cartan domain of genus p and rank T . Employing the standard notation (see [UU] ,[AZ],[FK2]) we denote by K ( z , w ) the Bergman kernel of R, p = ( P I , p 2 , . . . , pr) E (a*)@ N C the half-sum of the positive roots, and let 4A, A E (a*)@, be the usual labelling of the spherical functions:
where A(g) denotes the element of a such that exp(A(g)) is the A component of g under the Iwasawa decomposition G = N A K . For any L E DiffG(R), an invariant 1991 Mathematics Subject Classification.