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

• Autorzy: Marian Nowak
• Języki publikacji: EN

Abstrakty

EN

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.

Kategorie tematyczne

• 46E40: Spaces of vector- and operator-valued functions
• 47B38: Operators on function spaces (general)
• 46A20: Duality theory
• 47B07: Operators defined by compactness properties

• Wydawca: Institute of Mathematics Polish Academy of Sciences
• Czasopismo: Banach Center Publications
• Rocznik: 2005
• Tom: 68
• Numer: 1
• Strony: 109-114

Daty

wydano

2005

Twórcy

autor

Marian Nowak

• Faculty of Mathematics, Computer Science and Econometrics, University of Zielona Góra, Szafrana 4A, 65-516 Zielona Góra, Poland

Identyfikatory

DOI

10.4064/bc68-0-13