Homogeneous Tuples of Multiplication Operators on Twisted Bergman Spaces

AbstractLetBthe Bergman kernel on the domainΩn,mofn×mcontractive complex matrices (m⩾n⩾1). Let W=Wn,mbe the associated Wallach set consisting of theλ⩾0 for whichBλ/(m+n)is (non-negative definite and hence) the reproducing kernel of a functional Hilbert space H(λ). Forλ∈W, we examine themn-tupleM(λ)of operators on H(λ)whose components are multiplications by themnco-ordinate functions. This tuple is homogeneous with respect to the group action ofPSU(n, m) on the matrix ball. Utilising this group action we are able to determine the set of allλ∈W for which (i)M(λ)is bounded, and for which (ii)M(λ)is (bounded and) jointly subnormal. Further, the joint Taylor spectrum ofM(λ)is determined for allλas in (i). The subnormality ofM(λ)turns out to be closely tied with the representation theory ofPSU(n, m). Namely,M(λ)is subnormal precisely when the natural (projective) representation ofPSU(n, m) on the twisted Bergman space H(λ)is a subrepresentation of an induced representation of multiplicity 1. Finally, we examine the values ofλfor whichM(λ)admits its Taylor spectrum as ak-spectral set, and obtain incomplete results on this question. This question remains open and interesting onn−1 gaps, that is, forλbelonging to the union ofn−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 domainsΩn, mfor concreteness