Compact Toeplitz operators via the Berezin transform on bounded symmetric domains

Let Omega be an irreducible bounded symmetric domain of genus p, h(x,y) its Jordan triple determinant, and A(nu)(2)(Omega) the standard weighted Bergman space of holomorphic functions on Omega square-integrable with respect to the measure h(z,z)(v-p)dz. Extending the recent result of Axler and Zheng for Omega = D, nu = p =2 (the unweighted Bergman spate on the unit disc), we show that if S is a fmite sum of finite products of Toeplitz operators on A(nu)(2)(Omega) and nu is sufficiently large, then S is compact if and only if the Berezin transform (S) over tilde of S tends to zero as z approaches partial derivative Omega. An analogous assertion for the Fock space is also obtained.