Some spectral properties for operators acting on Rigged Hilbert spaces

2015

Operators on Rigged Hilbert spaces have been considered from the 80s of the 20th century on as good ones for describing several physical models whose observable set didn’t turn out to be a C∗-algebra.A notion of resolvent set for an operator acting in a rigged Hilbert space \(\mathcal{D}\subset \mathcal{H}\subset \mathcal{D}^{\times }\) is proposed. This set depends on a family of intermediate locally convex spaces living between \(\mathcal{D}\) and \(\mathcal{D}^{\times }\), called interspaces. Some properties of the resolvent set and of the corresponding multivalued resolvent function are derived and some examples are discussed.

Bounded elements of C*-inductive locally convex spaces

2013

The notion of bounded element of C*-inductive locally convex spaces (or C*-inductive partial *-algebras) is introduced and discussed in two ways: The first one takes into account the inductive structure provided by certain families of C*-algebras; the second one is linked to the natural order of these spaces. A particular attention is devoted to the relevant instance provided by the space of continuous linear maps acting in a rigged Hilbert space.

Singular Perturbations and Operators in Rigged Hilbert Spaces

2015

A notion of regularity and singularity for a special class of operators acting in a rigged Hilbert space \({\mathcal{D} \subset \mathcal{H}\subset \mathcal{D}^\times}\) is proposed and it is shown that each operator decomposes into a sum of a regular and a singular part. This property is strictly related to the corresponding notion for sesquilinear forms. A particular attention is devoted to those operators that are neither regular nor singular, pointing out that a part of them can be seen as perturbation of a self-adjoint operator on \({\mathcal{H}}\). Some properties for such operators are derived and some examples are discussed.

Operators in rigged Hilbert spaces: toward a spectral analysis

Representations of Quasi–local quasi *–algebras and non–commutative integration

2013

In this paper we are going to continue the analysis undertaken in [1] and [2] about the investigation on Quasi-local quasi *-algebras and their structure. Our aim is to show that any *-semisimple Quasi-local quasi *-algebra (A,A0) can be represented as a class of non-commutative L1-spaces.

Operators in Rigged Hilbert spaces: some spectral properties

2014

A notion of resolvent set for an operator acting in a rigged Hilbert space $\D \subset \H\subset \D^\times$ is proposed. This set depends on a family of intermediate locally convex spaces living between $\D$ and $\D^\times$, called interspaces. Some properties of the resolvent set and of the corresponding multivalued resolvent function are derived and some examples are discussed.

Some representation theorems for sesquilinear forms

2016

The possibility of getting a Radon-Nikodym type theorem and a Lebesgue-like decomposition for a non necessarily positive sesquilinear $\Omega$ form defined on a vector space $\mathcal D$, with respect to a given positive form $\Theta$ defined on $\D$, is explored. The main result consists in showing that a sesquilinear form $\Omega$ is $\Theta$-regular, in the sense that it has a Radon-Nikodym type representation, if and only if it satisfies a sort Cauchy-Schwarz inequality whose right hand side is implemented by a positive sesquilinear form which is $\Theta$-absolutely continuous. In the particular case where $\Theta$ is an inner product in $\mathcal D$, this class of sesquilinear form cov…