The possibility of getting a Radon–Nikodym type theorem and a Lebesgue-like decomposition for a not necessarily positive sesquilinear Ω form defined on a vector space D, with respect to a given positive form Θ defined on D, is explored. The main result consists in showing that a sesquilinear form Ω is Θ-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 Θ-absolutely continuous. In the particular case where Θ is an inner product in D, this class of sesquilinear form covers all standard examples. In the case of a form defined on a dense subspace D of Hilbert space H we give a sufficient condition for the equality Ω(ξ,η)=〈Tξ|η〉, with T a closable operator, to hold on a dense subspace of H
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
A notion of regularity and singularity for a special class of operators acting in a rigged Hilbert space D⊂H⊂D× 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 H. Some properties for such operators are derived and some examples are discussed
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