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Item Open Access Corrigendum to “representations of ⁎-semigroups associated to invariant kernels with values adjointable operators” [Linear Algebra Appl. 486 (2015) 361–388](2020) Ay, Serdar; Gheondea, AurelianShow more We correct a lemma by adding the assumption that the ordered ⁎-space is Archimedean and show by counter-examples and examples that this is needed.Show more Item Open Access Dilations of doubly invariant kernels valued in topologically ordered *- spaces(Bilkent University, 2018-06) Ay, SerdarShow more An ordered *-space Z is a complex vector space with a conjugate linear involution *, and a strict cone Z+ consisting of self adjoint elements. A topologically ordered *-space is an ordered *-space with a locally convex topology compatible with its natural ordering. A VE (Vector Euclidean) space, in the sense of Loynes, is a complex vector space equipped with an inner product taking values in an ordered *-space, and a VH (Vector Hilbert) space, in the sense of Loynes, is a VE-space with its inner product valued in a complete topologically ordered *-space and such that its induced locally convex topology is complete. On the other hand, dilation type theorems are important results that often realize a map valued in a certain space as a part of some simpler elements on a bigger space. Dilation results today are of an extraordinary large diversity and it is a natural question whether most of them can be uni*ed under general theorems. We study dilations of weakly positive semide*nite kernels valued in (topologically) ordered *-spaces, which are invariant under left actions of *-semigroups and right actions of semigroups, called doubly invariant. We obtain VE and VHspaces linearisations of such kernels, and on equal foot, their reproducing kernel spaces, and operator representations of the acting semigroups. The main results are used to unify many of the known dilation theorems for invariant positive semide*nite kernels with operator values, also for kernels valued in certain algebras, as well as to obtain some new dilation type results, in the context of Hilbert C*-modules, locally Hilbert C*-modules and VH-spaces.Show more Item Open Access Invariant weakly positive semidefinite kernels with values in topologically ordered ∗-spaces(Instytut Matematyczny PAN, 2019) Ay, Serdar; Gheondea, AurelianShow more 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.Show more Item Open Access A measure disintegration approach to spectral multiplicity for normal operators(Bilkent University, 2012) Ay, SerdarShow more In this thesis we studied the notion of direct integral Hilbert spaces, first introduced by J. von Neumann, and the closely related notion of decomposable operators, as defined in Kadison and Ringrose [1997] and Abrahamse and Kriete [1973]. Examples which show that some of the most familiar spaces in analysis are direct integral Hilbert spaces are presented in detail. Then we give a careful treatment of the notion of disintegration of a probability measure on a locally compact separable metric space, and using the machinery we obtain, a proof of the Spectral Multiplicity Theorem for Normal Operators employing the notion of disintegration of measures is given, based on Abrahamse and Kriete [1973], Arveson [1976], Arveson [2002]. In Chapter 5 the notion of essential preimage is presented in the sense of the article Abrahamse and Kriete [1973], and its relation with the spectral multiplicity function is discussed.Show more