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

Halimeh Ardakani, S. M. Sadegh Modarres Mosadegh, S. Mohammad Moshtaghioun

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 subspace $\mathcal M$ of $K(E,F)$, $\mathcal M^*$ has the Schur property if and only if all point evaluations $\mathcal M_1(x)=\{Tx:T\in\mathcal M_1\}$ and $\tilde{\mathcal M_1}(y^*)=\{T^*y^*:T\in\mathcal M_1\}$ are relatively norm compact, where $x\in E$, $y^*\in F^*$ and $\mathcal M_1$ is the closed unit ball of $\mathcal M$.