Analytic models for commuting operator tuples on bounded symmetric domains |
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| Authors: | Jonathan Arazy Miroslav Englis |
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| Affiliation: | Department of Mathematics, University of Haifa, Haifa 31905, Israel ; MÚ AV CR, Zitná 25, 11567 Prague 1, Czech Republic |
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| Abstract: | For a domain  in  and a Hilbert space  of analytic functions on  which satisfies certain conditions, we characterize the commuting  -tuples  of operators on a separable Hilbert space  such that  is unitarily equivalent to the restriction of  to an invariant subspace, where  is the operator  -tuple  on the Hilbert space tensor product  . For  the unit disc and  the Hardy space  , this reduces to a well-known theorem of Sz.-Nagy and Foias; for  a reproducing kernel Hilbert space on  such that the reciprocal  of its reproducing kernel is a polynomial in  and  , this is a recent result of Ambrozie, Müller and the second author. In this paper, we extend the latter result by treating spaces  for which  ceases to be a polynomial, or even has a pole: namely, the standard weighted Bergman spaces (or, rather, their analytic continuation)  on a Cartan domain corresponding to the parameter  in the continuous Wallach set, and reproducing kernel Hilbert spaces  for which  is a rational function. Further, we treat also the more general problem when the operator  is replaced by  ,  being a certain generalization of a unitary operator tuple. For the case of the spaces  on Cartan domains, our results are based on an analysis of the homogeneous multiplication operators on  , which seems to be of an independent interest. |
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| Keywords: | Coanalytic models reproducing kernels bounded symmetric domains commuting operator tuples functional calculus |
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