Order-weakly compact operators from vector-valued function spaces to Banach spaces

Abstract

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 } ) \,T(\{f\in E(X):\; \|f(\cdot )\|_{_X}\le 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.