Ideal properties of the Dunford integration operator

Let F be a Banach space. We establish necessary and sufficient conditions for the Dunford integration operator, from the space of F‐valued Dunford integrable functions to the bidual of F, to belong to a given operator ideal. We also show how this fact can be used to characterize important classes of Banach spaces, such as Banach spaces with the Banach‐Saks property, separable Banach spaces not containing c₀, Banach spaces not containing c₀ or ?₁ and Asplund spaces not containing c₀.