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Some Representation Theorems for Sesquilinear Forms

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 $Ω(ξ,η)=\langle{Tξ}|η\rangle$, with $T$ a closable operator, to hold on a dense subspace of $\mathcal H$.