摘要：Using the hierarchy of weakly null sequences introduced in [2], we introduce two newfamilies of operator classes. The first family simultaneously generalizes the completely continuousoperators and the weak Banach-Saks operators. The second family generalizes the class DP. Westudy the distinctness of these classes, and prove that each class is an operator ideal. We alsoinvestigate the properties possessed by each class, such as injectivity, surjectivity, and identificationof the dual class. We produce a number of examples, including the higher ordinal Schreier andBaernstein spaces. We prove ordinal analogues of several known results for Banach spaces withthe Dunford-Pettis, hereditary Dunford-Pettis property, and hereditary by quotients Dunford-Pettisproperty. For example, we prove that for any 0 ≤ ξ, ζ < ω1, a Banach space X has the hereditaryωξ, ωζ-Dunford Pettis property if and only if every seminormalized, weakly null sequence either hasa subsequence which is an `ωξ1-spreading model or a cωζ0-spreading model.
