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Invariant weakly positive semidefinite kernels with values in topologically ordered $*$-spaces

Tom 248 / 2019

Serdar Ay, Aurelian Gheondea Studia Mathematica 248 (2019), 255-294 MSC: Primary 47A20; Secondary 43A35, 46E22, 46L89. DOI: 10.4064/sm8807-1-2018 Opublikowany online: 8 April 2019

Streszczenie

We consider weakly positive semidefinite kernels valued in ordered $*$-spaces with or without certain topological properties, and investigate their linearisations (Kolmogorov decompositions) as well as their reproducing kernel spaces. The spaces of realisations are of VE (Vector Euclidean) or VH (Vector Hilbert) type, more precisely, vector spaces that possess gramians (vector valued inner products). The main results refer to the case when the kernels are invariant under certain actions of $*$-semigroups and show under which conditions $*$-representations on VE-spaces, or VH-spaces in the topological case, can be obtained. Finally, we show that these results unify most of dilation type results for invariant positive semidefinite kernels with operator values as well as recent results on positive semidefinite maps on $*$-semigroups with values operators from a locally bounded topological vector space to its conjugate $Z$-dual space, for $Z$ an ordered $*$-space.

Autorzy

  • Serdar AyDepartment of Mathematics
    Bilkent University
    06800 Bilkent, Ankara, Turkey
    e-mail
  • Aurelian GheondeaDepartment of Mathematics
    Bilkent University
    06800 Bilkent, Ankara, Turkey
    and
    Institutul de Matematică al Academiei Române
    C.P. 1-764, 014700 Bucureşti, România
    e-mail

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