Nuclear operators on Banach function spaces

Abstract

Let X be a Banach space and E be a perfect Banach function space over a finite measure space (Omega ,Sigma ,lambda ) such that L^infty subset Esubset L^1. Let E' denote the Köthe dual of E and tau (E,E') stand for the natural Mackey topology on E. It is shown that every nuclear operator T:Erightarrow X between the locally convex space (E,tau (E,E')) and a Banach space X is Bochner representable. In particular, we obtain that a linear operator T:L^infty rightarrow X between the locally convex space (L^infty ,tau (L^infty ,L^1)) and a Banach space X is nuclear if and only if its representing measure m_T:Sigma rightarrow X has the Radon-Nikodym property and |m_T|(Omega )=Vert TVert _{nuc} (= the nuclear norm of T). As an application, it is shown that some natural kernel operators on L^infty are nuclear. Moreover, it is shown that every nuclear operator T:L^infty rightarrow X admits a factorization through some Orlicz space L^varphi , that is, T=Scirc i_infty , where S:L^varphi rightarrow X is a Bochner representable and compact operator and i_infty :L^infty rightarrow L^varphi is the inclusion map.