Some Representation Theorems for Sesquilinear Forms

Abstract

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 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 $\Omega (\xi ,\eta )=⟨T\xi |\eta ⟩$, with $T$ a closable operator, to hold on a dense subspace of $\mathcal{H}$.

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