2016/07/21 by Di Bella, Salvatore, Trapani, Camillo
#FOS: Mathematics #Functional Analysis (math.FA)
paper · doi:10.48550/arxiv.1607.06216
The possibility of getting a Radon-Nikodym type theorem and a Lebesgue-like decomposition for a non necessarily positive sesquilinear Ω form defined on a vector space \mathcal 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 \mathcal D, this class of sesquilinear form covers all standard examples. In the case of a form defined on a dense subspace \mathcal D of Hilbert space \mathcal 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 \mathcal H.