Browsing by Subject "Hermitian kernel"

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    Dilations of some VH-spaces operator valued invariant Kernels
    (Springer, 2012) Gheondea, A.
    We investigate VH-spaces (Vector Hilbert spaces, or Loynes spaces) operator valued Hermitian kernels that are invariant under actions of *-semigroups from the point of view of generation of *-representations, linearizations (Kolmogorov decompositions), and reproducing kernel spaces. We obtain a general dilation theorem in both Kolmogorov and reproducing kernel space representations, that unifies many dilation results, in particular B. Sz.-Nagy's and Stinesprings' dilation type theorems. © 2012 Springer Basel.
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    Invariant weakly positive semidefinite kernels with values in topologically ordered ∗-spaces
    (Instytut Matematyczny PAN, 2019) Ay, Serdar; Gheondea, Aurelian
    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.