Abstract
We introduce generalized Schur functions and generalized positive functions in the setting of slice hyperholomorphic functions and study their realizations in terms of associated reproducing kernel Pontryagin spaces. To this end, we also prove some results in quaternionic functional analysis like an invariant subspace theorem for contractions in a Pontryagin space. We also consider slice hyperholomorphic functions on the half space \({\mathbb{H}_{+}}\) of quaternions with positive real parts and we study the Hardy space H 2 \({(\mathbb{H}_{+})}\) and Blaschke products in this framework.
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D. Alpay thanks the Earl Katz family for endowing the chair which supported his research, and the Binational Science Foundation Grant number 2010117. F. Colombo and I. Sabadini acknowledge the Center for Advanced Studies of the Mathematical Department of the Ben-Gurion University of the Negev for the support and the kind hospitality during the period in which part of this paper has been written.
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Alpay, D., Colombo, F., Lewkowicz, I. et al. Realizations of Slice Hyperholomorphic Generalized Contractive and Positive Functions. Milan J. Math. 83, 91–144 (2015). https://doi.org/10.1007/s00032-014-0231-9
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DOI: https://doi.org/10.1007/s00032-014-0231-9