The $ξ,ζ$-Dunford Pettis property

Abstract: Using the hierarchy of weakly null sequences introduced by Argyros, Merkourakis, and Tsarpalias, we introduce two new families of operator classes. The first family simultaneously generalizes the completely continuous operators and the weak Banach-Saks operators. The second family generalizes the class $\mathfrak{DP}$. We study the distinctness of these classes, and prove that each class is an operator ideal. We also investigate the properties possessed by each class, such as injectivity, surjectivity, and identification of the dual class. We produce a number of examples, including the higher ordinal Schreier and Baernstein spaces. We prove ordinal analogues of several known results for Banach spaces with the Dunford-Pettis, hereditary Dunford-Pettis property, and hereditary by quotients Dunford-Pettis property. For example, we prove that for any $0\leqslant \xi, \zeta<\omega_1$, a Banach space $X$ has the hereditary $\omega^\xi, \omega^\zeta$-Dunford Pettis property if and only if every seminormalized, weakly null sequence either has a subsequence which is an $\ell_1^{\omega^\xi}$-spreading model or a $c_0^{\omega^\zeta}$-spreading model.