Orderweakly compact operators from vectorvalued function spaces to Banach spaces
Author:
Marian Nowak
Journal:
Proc. Amer. Math. Soc. 135 (2007), 28032809
MSC (2000):
Primary 47B38, 47B07, 46E40, 46A20
Published electronically:
May 4, 2007
MathSciNet review:
2317955
Fulltext PDF Free Access
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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 orderweakly compact whenever for each the set is relatively weakly compact in . In this paper we examine orderweakly compact operators . We give a characterization of an orderweakly 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 orderweakly compact. Moreover, it is proved that if is a Banach function space, then for every Banach space any continuous linear operator is orderweakly compact iff the norm is order continuous and is reflexive. In particular, for every Banach space any continuous linear operator is orderweakly 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/S0002993907088284
PII:
S 00029939(07)088284
Keywords:
Vectorvalued function spaces,
K\"otheBochner spaces,
orderbounded operators,
orderweakly 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
