Order-weakly Compact Operators from Vector-valued Function Spaces to Banach Spaces

Let E be an ideal of L0 over a σ-finite measure space (Ω,Σ, μ), and let E∼ stand for the order dual of E. For a real Banach space (X, ‖ · ‖X ) let E(X) be a subspace of the space L0(X) of μ-equivalence classes of strongly Σ-measurable functions f : Ω −→ X and consisting of all those f ∈ L0(X) for which the scalar function ‖f(·)‖X belongs to E. For a real Banach space (Y, ‖ · ‖Y ) a linear operator T : E(X) −→ Y is said to be order-weakly compact whenever for each u ∈ E+ the set T ({f ∈ E(X) : ‖f(·)‖X ≤ u}) is relatively weakly compact in Y . In this paper we examine order-weakly compact operators T : E(X) −→ Y . We give a characterization of an order-weakly compact operator T in terms of the continuity of the conjugate operator of T with respect to some weak topologies. It is shown that if (E, ‖ · ‖E ) is an order continuous Banach function space, X is a Banach space containing no isomorphic copy of l1 and Y is a weakly sequentially complete Banach space, then every continuous linear operator T : E(X) −→ Y is order-weakly compact. Moreover, it is proved that if (E, ‖ · ‖E ) is a Banach function space, then for every Banach space Y any continuous linear operator T : E(X) −→ Y is order-weakly compact iff the norm ‖ · ‖E is order continuous and X is reflexive. In particular, for every Banach space Y any continuous linear operator T : L1(X) −→ Y is order-weakly compact iff X is reflexive.