Yosida-Hewitt type decompositions for order-weakly compact operators

Abstract

Let $E$ be an ideal of $L^0$ over a $\Sigma$-finite measure space $\,(\Omega,\Sigma,\mu)$. For a real Banach space $\,(X,\|\cdot\|_X)\,$ let $\,E(X)\,$ be the subspace of the space $L^0(X)$ of $\mu$-equivalence classes of strongly $\,\Sigma$-measurable functions $\,f:\Omega\rightarrow X\,$ consisting of all those $f\in L^0(X)$ for which the scalar function $\,\|f(\cdot)\|_X\,$ belongs to $\,E$. For a real Banach space $\,Y\,$ a linear operator $\,T:E(X)\rightarrow Y\,$ is said to be order-weakly compact whenever for each $\,u\in E^+$ the set $\,T(\{f\in E(X):\|f(\cdot)\|_X\le u\})\,$ is relatively weakly compact in $\,Y$. In this paper we derive Yosida-Hewitt type decompositions for order-weakly compact operators $\,T:E(X)\rightarrow Y$. In particular, it is shown that if $\,X\,$ is an Asplund space, then an order-weakly compact operator $\,T:E(X)\rightarrow Y\,$ can be uniquely decomposed as $\,T=T_1+T_2$, where $\,T_1, T_2$ are order-weakly compact operators, $\,T_1$ is smooth and $\,T_2$ is weakly singular.