Bulletin of the
Korean Mathematical Society BKMS

For several Banach spaces $X$ and $Y$ and operator ideal ${\cal U}$, if ${\cal U}(X,Y)$ denotes the component of operator ideal ${\cal U}$; according to Freedman's definitions, it is shown that a necessary and sufficient condition for a closed subspace ${\cal M}$ of ${\cal U}(X,Y)$ to have the alternative Dunford-Pettis property is that all evaluation operators $\phi_x:{\cal M}\to Y$ and $\psi_{y^*}:{\cal M}\to X^*$ are DP1 operators, where $\phi_x(T)= Tx$ and $\psi_{y^*}(T)= T^*y^*$ for $x\in X$, $y^*\in Y^*$ and $T\in {\cal M}.$

Keywords: Dunford-Pettis property, Schauder decomposition, compact operator, operator ideal

MSC numbers: Primary 47L05, 47L20; Secondary 46B28, 46B99