Analytic models for commuting operator tuples on bounded symmetric domains

Abstract

For a domainΩ\OmegainCd\mathbb {C}^{d}and a Hilbert spaceH\mathcal {H}of analytic functions onΩ\Omegawhich satisfies certain conditions, we characterize the commutingdd-tuplesT=(T1,…,Td)T=(T_{1},\dots ,T_{d})of operators on a separable Hilbert space HHsuch thatT∗T^{*}is unitarily equivalent to the restriction ofM∗M^{*}to an invariant subspace, whereMMis the operatordd-tupleZ⊗IZ\otimes Ion the Hilbert space tensor product H⊗H\mathcal {H} \otimes H. ForΩ\Omegathe unit disc andH\mathcal {H}the Hardy space H2H^{2}, this reduces to a well-known theorem of Sz.-Nagy and Foias; forH\mathcal {H}a reproducing kernel Hilbert space onΩ⊂Cd\Omega \subset \mathbb {C} ^{d}such that the reciprocal1/K(x,y¯)1/K(x,\overline {y})of its reproducing kernel is a polynomial inxxand y¯\overline y, this is a recent result of Ambrozie, Müller and the second author. In this paper, we extend the latter result by treating spacesH\mathcal {H}for which1/K1/Kceases to be a polynomial, or even has a pole: namely, the standard weighted Bergman spaces (or, rather, their analytic continuation)H=Hν\mathcal {H} =\mathcal {H} _{\nu }on a Cartan domain corresponding to the parameterν\nuin the continuous Wallach set, and reproducing kernel Hilbert spacesH\mathcal {H}for which1/K1/Kis a rational function. Further, we treat also the more general problem when the operatorMMis replaced by M⊕WM\oplus W,WW being a certain generalization of a unitary operator tuple. For the case of the spacesHν\mathcal {H} _{\nu }on Cartan domains, our results are based on an analysis of the homogeneous multiplication operators onΩ\Omega, which seems to be of an independent interest.