Homogeneous multiplication operators on bounded symmetric domains

Let D=G/K be an irreducible bounded symmetric domain of dimension d and let Hν(D) be the analytic continuation of the weighted Bergman spaces of holomorphic functions on D. We consider the d-tuple M=(M1,…,Md) of multiplication operators by coordinate functions and consider its spectral properties. We find those parameters ν for which the tuple M is subnormal and we answer some open questions of Bagchi and Misra. In particular, we prove that when D=Bd is the unit ball in Cd, then Bd is a k-spectral set of M if and only if Hν(Bd) is the Hardy space or a weighted Bergman space.