Homogeneous multiplication operators on bounded symmetric domains

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