arXiv subjects

Aurelian Gheondea

Publications and source records attributed to Aurelian Gheondea.

21 records · Page 2Linked to original sources

arXiv2013

A Survey on Reproducing Kernel Krein Spaces

This is a survey on reproducing kernel Krein spaces and their interplay with operator valued Hermitian kernels. Existence and uniqueness properties are carefully reviewed. The approach we follow in this survey uses a more abstract but very useful concept of linearization or Kolmogorov decomposition, as well as the underlying concept of Krein space induced by a selfadjoint operator and that of Krein space continuously embedded. The operator range feature of reproducing kernel spaces is emphasized. We include a careful presentation of Hermitian kernels on complex regions that points out a universality property of the Szego kernels with respect to reproducing kernel Krein spaces of holomorphic functions.

math.FA↗

arXiv2013

An Interpolation Problem for Completely Positive Maps on Matrix Algebras: Solvability and Parametrisation

We present certain existence criteria and parameterisations for an interpolation problem for completely positive maps that take given matrices from a finite set into prescribed matrices. Our approach uses density matrices associated to linear functionals on *-subspaces of matrices, inspired by the Smith-Ward linear functional and Arveson's Hahn-Banach type Theorem. We perform a careful investigation on the intricate relation between the positivity of the density matrix and the positivity of the corresponding linear functional. A necessary and sufficient condition for the existence of solutions and a parametrisation of the set of all solutions of the interpolation problem in terms of a closed and convex set of an affine space are obtained. Other linear affine restrictions, like trace preserving, can be included as well, hence covering applications to quantum channels that yield certain quantum states at prescribed quantum states.

math.OA↗

arXiv2000

Representations of hermitian kernels by means of Krein spaces II. Invariant kernels

In this paper we study hermitian kernels invariant under the action of a semigroup with involution. We characterize those hermitian kernels which realize the given action by bounded operators on a Krein space. Applications to the GNS representation of *-algebras associated to hermitian functionals are given. We explain the key role played by the Kolmogorov decomposition in the construction of Weyl exponentials associated to an indefinite inner product and in the dilation thoery of hermitian maps on C*-algebras.

math.FA↗