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Journal of the London Mathematical Society Advance Access originally published online on March 24, 2009
Journal of the London Mathematical Society 2009 79(3):612-630; doi:10.1112/jlms/jdp007
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© 2009 London Mathematical Society

Characterizations of strictly singular operators on Banach lattices

J. Flores

Department of Applied Mathematics Escet
Universidad Rey Juan Carlos
Móstoles 28933
Madrid
Spain
julio.flores@urjc.es

F. L. Hernández

Departamento de Análisis Matemático
Universidad Complutense de Madrid
28040 Madrid
Spain
pacoh@mat.ucm.es

N. J. Kalton

Department of Mathematics
University of Missouri
Columbia, MO 65211
USA
nigel@math.missouri.edu

P. Tradacete

Departamento de Análisis Matemático
Universidad Complutense de Madrid
28040 Madrid
Spain

New characterizations of strictly singular operators between Banach lattices are given. It is proved that, for Banach lattices X and Y such that X has finite cotype and Y satisfies a lower 2-estimate, an operator T : X -> Y is strictly singular if and only if it is disjointly strictly singular and {ell}2-singular. Moreover, if T is regular then the same equivalence holds provided that Y is just order continuous. Furthermore, it is shown that these results fail if the conditions on the lattices are relaxed.

2000 Mathematics Subject Classification 46B42, 47B07, 47B60.

The first, second, and fourth authors were partially supported by Spanish grants MTM2005-00082, UCM910346, and PR34/07-15837. The third author was supported by NSF grant DMS-0555670. The fourth author was partially supported by MEC grant AP-2004-4841.

Received July 8, 2008; published online March 24, 2009.


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