Strongly alternative Dunford - Pettis subspaces of operator ideals

Abstract

Introducing the concept of strong alternative Dunford – Pettis property (strong DP1) for the subspace M of operator ideals $\mathcal{U}(X, Y )$ between Banach spaces $X$ and $Y$, we show that M is a strong DP1 subspace if and only if all evaluation operators $\phi_x : \mathcal{M} → Y$ та $ψy∗ : \mathcal{M} → X^{*}$ are DP1 operators, where $\phi_x(T) = T x$ та $ψ_{y^{∗}} (T) = T^{∗}y^{∗}$ for $x ∈ X, y^{∗} ∈ Y$ and $T ∈ M$. Some consequences related to the concept of alternative Dunford – Pettis property in subspaces of some operator ideals are obtained.