Order-bounded operators from vector-valued function spaces to Banach spaces

Marian Nowak

Banach Center Publications, Tome 68 (2005), p. 109-114 / Harvested from The Polish Digital Mathematics Library

Résumé

Let E be an ideal of L⁰ over a σ-finite measure space (Ω,Σ,μ). 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. Let E(X)˜ stand for the order dual of E(X). For u ∈ E⁺ let $D_u ( = {f ∈ E(X): ||f(·)||_X ≤ u})$ stand for the order interval in E(X). For a real Banach space $(Y,||·||_Y)$ a linear operator T: E(X) → Y is said to be order-bounded whenever for each u ∈ E⁺ the set $T(D_u)$ is norm-bounded in Y. In this paper we examine order-bounded operators T: E(X) → Y. We show that T is order-bounded iff T is $(τ(E(X),E(X)˜),||·||_Y)$-continuous. We obtain that every weak Dunford-Pettis operator T: E(X) → Y is order-bounded. In particular, we obtain that if a Banach space Y has the Dunford-Pettis property, then T is order-bounded iff it is a weak Dunford-Pettis operator.

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Publié le : 2005-01-01

Zbl 1076.47023

EUDML-ID : urn:eudml:doc:282142