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Proceedings of the American Mathematical Society
Proceedings of the American Mathematical Society
ISSN 1088-6826(online) ISSN 0002-9939(print)

 

Order-weakly compact operators from vector-valued function spaces to Banach spaces


Author: Marian Nowak
Journal: Proc. Amer. Math. Soc. 135 (2007), 2803-2809
MSC (2000): Primary 47B38, 47B07, 46E40, 46A20
Published electronically: May 4, 2007
MathSciNet review: 2317955
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Abstract: Let $ E$ be an ideal of $ L^{0}$ over a  $ \sigma$-finite measure space $ ( \Omega, \Sigma,\mu)$, and let $ E^{\sim}$ stand for the order dual of $ E$. For a real Banach space $ (X,\Vert\cdot\Vert _{_X})$ let $ E(X)$ be a subspace of the space $ L^{0}(X)$ of $ \mu$-equivalence classes of strongly $ \Sigma$-measurable functions $ f : \Omega \longrightarrow X$ and consisting of all those $ f \in L^{0}(X)$ for which the scalar function $ \Vert f(\cdot)\Vert _{_X}$ belongs to $ E$. For a real Banach space $ (Y, \Vert\cdot\Vert _{_Y})$ a linear operator $ T : E(X) \longrightarrow Y$ is said to be order-weakly compact whenever for each $ u \in E^+$ the set $ \,T(\{f\in E(X):\; \Vert f(\cdot)\Vert _{_X}\le u\})\,$ is relatively weakly compact in $ Y$. In this paper we examine order-weakly compact operators $ T : E(X) \longrightarrow 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,\Vert\cdot\Vert _{_E})$ is an order continuous Banach function space, $ X$ is a Banach space containing no isomorphic copy of $ l^{1}$ and $ Y$ is a weakly sequentially complete Banach space, then every continuous linear operator $ T : E(X) \longrightarrow Y$ is order-weakly compact. Moreover, it is proved that if $ (E,\Vert\cdot\Vert _{_E})$ is a Banach function space, then for every Banach space $ Y$ any continuous linear operator $ T : E(X) \longrightarrow Y$ is order-weakly compact iff the norm $ \Vert\cdot\Vert _{_E}$ is order continuous and $ X$ is reflexive. In particular, for every Banach space $ Y$ any continuous linear operator $ T : L^{1}(X) \longrightarrow Y$ is order-weakly compact iff $ X$ is reflexive.


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Additional Information

Marian Nowak
Affiliation: Faculty of Mathematics, Computer Science and Econometrics, University of Zielona Góra, ul. Szafrana 4A, 65–001 Zielona Góra, Poland
Email: M.Nowak@wmie.uz.zgora.pl

DOI: http://dx.doi.org/10.1090/S0002-9939-07-08828-4
PII: S 0002-9939(07)08828-4
Keywords: Vector-valued function spaces, K\"othe-Bochner spaces, order-bounded operators, order-weakly compact operators, order intervals.
Received by editor(s): December 18, 2003
Received by editor(s) in revised form: May 18, 2006
Published electronically: May 4, 2007
Communicated by: Jonathan M. Borwein
Article copyright: © Copyright 2007 American Mathematical Society