### 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 originale | English |
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pagine (da-a) | 64-83 |

Numero di pagine | 20 |

Rivista | Journal of Mathematical Analysis and Applications |

Volume | 451 |

Stato di pubblicazione | Published - 2017 |

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### All Science Journal Classification (ASJC) codes

- Analysis
- Applied Mathematics