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

Marian 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 -