A Radon-Nikodym theorem for nonlinear functionals on Banach lattices

Feldman, William

doi:10.1090/bproc/128

A Radon-Nikodym theorem for nonlinear functionals on Banach lattices
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by William Feldman HTML | PDF

Proc. Amer. Math. Soc. Ser. B 9 (2022), 150-158

Abstract:

A Radon-Nikodym theorem is established for a class of nonlinear orthogonally additive monotone functionals on Dedekind complete Banach lattices. A functional $S$ is absolutely continuous with respect to $T$ if $T(f) =0$ implies $S( f)=0$ for $f$ in the domain. It is shown that $S$ is absolutely continuous with respect to $T$ implies $S$ is equal to the composition of an extension of $T$ with an appropriate generalized orthomorphism. In the special case that $S$ and $T$ are linear, the generalized orthomorphism reduces to a multiplication operator consistent with the classical formulation of this theorem.

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