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.Comment: 31 page

Triplets of Closely Embedded Hilbert Spaces

We obtain a general concept of triplet of Hilbert spaces with closed (unbounded) embeddings instead of continuous (bounded) ones. The construction starts with a positive selfadjoint operator $H$, that is called the Hamiltonian of the system, which is supposed to be one-to-one but may not have a bounded inverse, and for which a model is obtained. From this model we get the abstract concept and show that its basic properties are the same with those of the model. Existence and uniqueness results, as well as left-right symmetry, for these triplets of closely embedded Hilbert spaces are obtained. We motivate this abstract theory by a diversity of problems coming from homogeneous or weighted Sobolev spaces, Hilbert spaces of holomorphic functions, and weighted $L^2$ spaces. An application to weak solutions for a Dirichlet problem associated to a class of degenerate elliptic partial differential equations is presented. In this way, we propose a general method of proving the existence of weak solutions that avoids coercivity conditions and Poincar\'e-Sobolev type inequalities.Comment: 29 page

Representations of ∗*-semigroups associated to invariant kernels with values adjointable operators. I

We consider positive semidefinite kernels valued in the $*$-algebra of adjointable operators on a VE-space (Vector Euclidean space) and that are invariant under actions of $*$-semigroups. A rather general dilation theorem is stated and proved: for these kind of kernels, representations of the $*$-semigroup on either the VE-spaces of linearisation of the kernels or on their reproducing kernel VE-spaces are obtainable. We point out the reproducing kernel fabric of dilation theory and we show that the general theorem unifies many dilation results at the non topological level.Comment: 23 page

Dilations of some VH-spaces operator valued invariant kernels

Cataloged from PDF version of article.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

Embeddings, operator ranges, and Dirac operators

Cataloged from PDF version of article.Motivated by energy space representation of Dirac operators, in the sense of K. Friedrichs, we recently introduced the notion of closely embedded Kreǐn spaces. These spaces are associated to unbounded selfadjoint operators that play the role of kernel operators, in the sense of L. Schwartz, and they are special representations of induced Kreǐn spaces. In this article we present a canonical representation of closely embedded Kreǐn spaces in terms of a generalization of the notion of operator range and obtain a characterization of uniqueness. When applied to Dirac operators, the results differ according to a mass or a massless particle in a dramatic way: in the case of a particle with a nontrivial mass we obtain a dual of a Sobolev type space and we have uniqueness, while in the case of a massless particle we obtain a dual of a homogenous Sobolev type space and we lose uniqueness. © 2010 Elsevier Inc