Homogeneous tuples of multiplication operators on twisted Bergman spaces

Abstract

Let B the Bergman kernel on the domain Ω_n,m of n×m contractive complex matrices (m ≥ n ≥ 1). Let W=W _n,m be the associated Wallach set consisting of the λ ≥ 0 for which β^λ/(m+n)is (non-negative definite and hence) the reproducing kernel of a functional Hilbert space H ^(λ). For λ ∈ W , we examine the mn-tuple M^(λ) of operators on H^(λ) whose components are multiplications by the mnco-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 λ ∈ W for which (i) M^(λ) is bounded, and for which (ii) M^(λ) is (bounded and) jointly subnormal. Further, the joint Taylor spectrum of M(λ) is determined for all λ as in (i). The subnormality of M^(λ) turns out to be closely tied with the representation theory of PSU(n, m). Namely,M^(λ) is subnormal precisely when the natural (projective) representation of PSU(n, m) on the twisted Bergman space H(λ) is a subrepresentation of an induced representation of multiplicity 1. Finally, we examine the values of λ for which M^(λ) 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 gamma 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 Ω_n,m for concreteness.