THE ALTERNATIVE DUNFORD-PETTIS PROPERTY IN SUBSPACES OF OPERATOR IDEALS

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