A Radon-Nikodym theorem for nonlinear functionals on Banach lattices

Abstract

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