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

Banach Center Publications (2005)

- Volume: 68, Issue: 1, page 109-114
- ISSN: 0137-6934

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topMarian Nowak. "Order-bounded operators from vector-valued function spaces to Banach spaces." Banach Center Publications 68.1 (2005): 109-114. <http://eudml.org/doc/282142>.

@article{MarianNowak2005,

abstract = {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.},

author = {Marian Nowak},

journal = {Banach Center Publications},

keywords = {vector-valued function spaces; Köthe–Bochner spaces; order-bounded operators; order intervals; Dunford–Pettis property; weak Dunford–Pettis operators},

language = {eng},

number = {1},

pages = {109-114},

title = {Order-bounded operators from vector-valued function spaces to Banach spaces},

url = {http://eudml.org/doc/282142},

volume = {68},

year = {2005},

}

TY - JOUR

AU - Marian Nowak

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

JO - Banach Center Publications

PY - 2005

VL - 68

IS - 1

SP - 109

EP - 114

AB - 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.

LA - eng

KW - vector-valued function spaces; Köthe–Bochner spaces; order-bounded operators; order intervals; Dunford–Pettis property; weak Dunford–Pettis operators

UR - http://eudml.org/doc/282142

ER -

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