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 be an ideal of over a -finite measure space , and let stand for the order dual of . For a real Banach space let be a subspace of the space of -equivalence classes of strongly -measurable functions and consisting of all those for which the scalar function belongs to . For a real Banach space a linear operator is said to be order-weakly compact whenever for each the set is relatively weakly compact in . In this paper we examine order-weakly compact operators . We give a characterization of an order-weakly compact operator in terms of the continuity of the conjugate operator of with respect to some weak topologies. It is shown that if is an order continuous Banach function space, is a Banach space containing no isomorphic copy of and is a weakly sequentially complete Banach space, then every continuous linear operator is order-weakly compact. Moreover, it is proved that if is a Banach function space, then for every Banach space any continuous linear operator is order-weakly compact iff the norm is order continuous and is reflexive. In particular, for every Banach space any continuous linear operator is order-weakly compact iff 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

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