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.

Di Bella, S., & Trapani, C. (2017). Some representation theorems for sesquilinear forms. JOURNAL OF MATHEMATICAL ANALYSIS AND APPLICATIONS, 451(1), 64-83 [10.1016/j.jmaa.2017.01.079].

Some representation theorems for sesquilinear forms

DI BELLA, Salvatore;TRAPANI, Camillo
2017

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.
https://www.sciencedirect.com/science/article/pii/S0022247X17301129?via=ihub
Di Bella, S., & Trapani, C. (2017). Some representation theorems for sesquilinear forms. JOURNAL OF MATHEMATICAL ANALYSIS AND APPLICATIONS, 451(1), 64-83 [10.1016/j.jmaa.2017.01.079].
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Utilizza questo identificativo per citare o creare un link a questo documento: http://hdl.handle.net/10447/225814
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