Pontryagin-de Branges-Rovnyak spaces of slice hyperholomorphic functions

Daniel Alpay, Fabrizio Colombo, Irene Sabadini

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66 Scopus citations

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.

Original languageEnglish
Pages (from-to)87-125
Number of pages39
JournalJournal d'Analyse Mathematique
Volume121
Issue number1
DOIs
StatePublished - 1 Oct 2013

ASJC Scopus subject areas

  • Analysis
  • General Mathematics

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