Some representation theorems for sesquilinear forms

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Abstract

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.
Lingua originaleEnglish
pagine (da-a)64-83
Numero di pagine20
RivistaJournal of Mathematical Analysis and Applications
Volume451
Stato di pubblicazionePublished - 2017

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All Science Journal Classification (ASJC) codes

  • Analysis
  • Applied Mathematics

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