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