Let B the Bergman kernel on the domain Omega(n,m) of n x m contractive complex matrices (m greater than or equal to n greater than or equal to 1). Let W-n,W-m be the associated Wallach set consisting of the lambda greater than or equal to 0 for which B-lambda/(m+n) is (non-negative definite and hence) the reproducing kernel of a functional Hilbert space H-(lambda). For lambda is an element of W, we examine the mn-tuple M((lambda)) of operators on H-(lambda) whose components are multiplications by the mit co-ordinate functions. This tuple is homogeneous with respect to the group action of PSU(n, m) on the matrix ball. Utilising this group action we are able to determine the set of all lambda is an element of W for which (i) M((lambda)) is bounded, and for which (ii) M((lambda)) is (bounded and) jointly subnormal. Further, the joint Taylor spectrum of M((lambda)) is determined for all lambda as in (i). The subnormality of M((lambda)) turns out to be closely tied with the representation theory of PSU(n, m). Namely, M((lambda)) is subnormal precisely when the natural (projective) representation of PSU(n, m) on the twisted Bergman space H-(lambda) is a subrepresentation of an induced representation of multiplicity 1. Finally, we examine the values of lambda for which M((lambda)) admits its Taylor spectrum as a k-spectral set, and obtain incomplete results on this question. This question remains open and interesting on n - 1 gaps, that is, for lambda belonging to the union of n - 1 pairwise disjoint open intervals. Most of the techniques developed in this paper are applicable to all bounded Cartan domains, though we stick to the matrix domains Omega(n, m) for concreteness. (C) 1996 Academic Press, Inc. [References: 16]
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