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Reproducing Kernel Kreĭn Spaces

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Operator Theory

Abstract

This chapter is an introduction to reproducing kernel Kreĭn spaces and their interplay with operator valued Hermitian kernels. Existence and uniqueness properties are carefully reviewed. The approach used in this survey involves the more abstract, but very useful, concept of linearization or Kolmogorov decomposition, as well as the underlying concepts of Kreĭn space induced by a selfadjoint operator and that of Kreĭn space continuously embedded. The operator range feature of reproducing kernel spaces is emphasized. A careful presentation of Hermitian kernels on complex regions that point out a universality property of the Szegö kernels with respect to reproducing kernel Kreĭn spaces of holomorphic functions is included.

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Acknowledgements

Work supported by a grant of the Romanian National Authority for Scientific Research, CNCS UEFISCDI, project number PN-II-ID-PCE-2011-3-0119.

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Authors and Affiliations

  1. Department of Mathematics and Institutul de Matematicăal Academiei, Bilkent University, Ankara, Turkey

    Aurelian Gheondea

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  1. Aurelian Gheondea
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Correspondence to Aurelian Gheondea .

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Editors and Affiliations

  1. Earl Katz Chair in Algebraic System Theory, Department of Mathematics, Ben-Gurion University of the Negev, Be’er Sheva, Israel

    Daniel Alpay

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Gheondea, A. (2015). Reproducing Kernel Kreĭn Spaces. In: Alpay, D. (eds) Operator Theory. Springer, Basel. https://doi.org/10.1007/978-3-0348-0667-1_40

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