Introducing the concept of strong alternative Dunford–Pettis property (strong DP1) for the subspace \( \mathcal{M} \) of operator ideals \( \mathcal{U} \)(X, Y) between Banach spaces X and Y, we show that \( \mathcal{M} \) is a strong DP1 subspace if and only if all evaluation operators \( {\phi_x}:\mathcal{M}\to Y \) and \( {\psi_{{{y^{*}}}}}:\mathcal{M}\to {X^{*}} \) are DP1 operators, where ϕ x (T) = Tx and ψ y* (T) = T*y* for x ∈ X, y* ∈ Y*, and T ∈ \( \mathcal{M} \). Some consequences related to the concept of alternative Dunford–Pettis property in the subspaces of some operator ideals are obtained.
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Published in Ukrains’kyi Matematychnyi Zhurnal, Vol. 65, No. 4, pp. 588–593, April, 2013.
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Moshtaghioun, S.M. Strongly alternative Dunford–Pettis subspaces of operator ideals. Ukr Math J 65, 649–655 (2013). https://doi.org/10.1007/s11253-013-0802-x
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DOI: https://doi.org/10.1007/s11253-013-0802-x