Weak Sequential Convergence in the Dual of Compact Operators between Banach Lattices

Abstract

For several Banach lattices $E$ and $F$, if $K(E,F)$ denotes the space of all compact operators from $E$ to $F$, under some conditions on $E$ and $F$, it is shown that for a closed sublattice $\mathcal{M}$ of $ K(E,F)$, $\mathcal{M}^*$ has the Schur property if and only if all evaluation operators $\phi_x: \mathcal{M} \rightarrow F$ and $\psi_{y^*} :\mathcal{M} \rightarrow E^*$ are compact operators, where $\phi_x (T)= Tx$ and $\psi_{y^*} (T)=T^*y^*$ for $x\in E$, $y^*\in F^*$ and $T\in \mathcal{M}$.