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Pontryagin-de Branges-Rovnyak spaces of slice hyperholomorphic functions

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Abstract

We study reproducing kernel Hilbert and Pontryagin spaces of slice hyperholomorphic functions. These are analogs of the Hilbert spaces of analytic functions introduced by de Branges and Rovnyak. In the first part of the paper, we focus on the case of Hilbert spaces and introduce, in particular, a version of the Hardy space. Then we define Blaschke factors and Blaschke products and consider an interpolation problem. In the second part of the paper, we turn to the case of Pontryagin spaces. We first prove some results from the theory of Pontryagin spaces in the quaternionic setting and, in particular, a theorem of Shmulyan on densely defined contractive linear relations. We then study realizations of generalized Schur functions and of generalized Carathéodory functions.

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Authors and Affiliations

  1. Department of Mathematics, Ben-Gurion University of the Negev, Beer-Sheva, 84105, Israel

    Daniel Alpay

  2. Politecnico di Milano, Dipartimento di Matematica, Via E. Bonardi, 9, 20133, Milano, Italy

    Fabrizio Colombo & Irene Sabadini

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  1. Daniel Alpay
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  2. Fabrizio Colombo
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  3. Irene Sabadini
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Correspondence to Daniel Alpay.

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Alpay, D., Colombo, F. & Sabadini, I. Pontryagin-de Branges-Rovnyak spaces of slice hyperholomorphic functions. JAMA 121, 87–125 (2013). https://doi.org/10.1007/s11854-013-0028-8

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  • DOI: https://doi.org/10.1007/s11854-013-0028-8

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