Positivity

, Volume 15, Issue 1, pp 161–174 | Cite as

Pontryagin space structure in reproducing kernel Hilbert spaces over *-semigroups

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Abstract

The geometry of spaces with indefinite inner product, known also as Krein spaces, is a basic tool for developing Operator Theory therein. In the present paper we establish a link between this geometry and the algebraic theory of *-semigroups. It goes via the positive definite functions and related to them reproducing kernel Hilbert spaces. Our concern is in describing properties of elements of the semigroup which determine shift operators which serve as Pontryagin fundamental symmetries.

Keywords

*-Semigroup Shift operator Pontryagin space Fundamental symmetry 

Mathematics Subject Classification (2000)

Primary 43A35 46C20 47B32 

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Copyright information

© Birkhäuser / Springer Basel AG 2010

Authors and Affiliations

  • Franciszek Hugon Szafraniec
    • 1
  • Michał Wojtylak
    • 1
  1. 1.Instytut MatematykiUniwersytet JagiellońskiKrakówPoland

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