Analytic models for commuting operator tuples on bounded symmetric domains

Abstract:

For a domain $\Omega$ in $\mathbb {C}^{d}$ and a Hilbert space $\mathcal {H}$ of analytic functions on $\Omega$ which satisfies certain conditions, we characterize the commuting $d$-tuples $T=(T_{1},\dots ,T_{d})$ of operators on a separable Hilbert space $H$ such that $T^{*}$ is unitarily equivalent to the restriction of $M^{*}$ to an invariant subspace, where $M$ is the operator $d$-tuple $Z\otimes I$ on the Hilbert space tensor product $\mathcal {H} \otimes H$. For $\Omega$ the unit disc and $\mathcal {H}$ the Hardy space $H^{2}$, this reduces to a well-known theorem of Sz.-Nagy and Foias; for $\mathcal {H}$ a reproducing kernel Hilbert space on $\Omega \subset \mathbb {C} ^{d}$ such that the reciprocal $1/K(x,\overline {y})$ of its reproducing kernel is a polynomial in $x$ and $\overline y$, this is a recent result of Ambrozie, Müller and the second author. In this paper, we extend the latter result by treating spaces $\mathcal {H}$ for which $1/K$ ceases to be a polynomial, or even has a pole: namely, the standard weighted Bergman spaces (or, rather, their analytic continuation) $\mathcal {H} =\mathcal {H} _{\nu }$ on a Cartan domain corresponding to the parameter $\nu$ in the continuous Wallach set, and reproducing kernel Hilbert spaces $\mathcal {H}$ for which $1/K$ is a rational function. Further, we treat also the more general problem when the operator $M$ is replaced by $M\oplus W$, $W$ being a certain generalization of a unitary operator tuple. For the case of the spaces $\mathcal {H} _{\nu }$ on Cartan domains, our results are based on an analysis of the homogeneous multiplication operators on $\Omega$, which seems to be of an independent interest.