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Bulletin of the London Mathematical Society Advance Access originally published online on May 21, 2008
Bulletin of the London Mathematical Society 2008 40(4):581-592; doi:10.1112/blms/bdn030
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© 2008 London Mathematical Society

Norm-attaining operators between Marcinkiewicz and Lorentz spaces

María D. Acosta

Departamento de Análisis Matemático
Universidad de Granada
18071 Granada
Spain
dacosta@ugr.es

Anna Kaminska

Department of Mathematical Sciences
The University of Memphis
Memphis, TN 38152
USA

Received 20 June 2007. Revision received 15 January 2008.

Bishop and Phelps proved that the set of norm-attaining functionals on any Banach space is dense in the topological dual. After that, the study of the same kind of problems for operators was initiated by Lindenstrauss, and several general positive results were proved. It was then consistently continued for different classes of spaces including L1(µ) or C(K). Here a similar problem is studied in the context of classical interpolation Marcinkiewicz and Lorentz spaces, MFormula and {Lambda}1, v, in both the real and the complex cases. We show that if wv isin L1 then the identity operator between these spaces is bounded, but it is not possible to approximate it by norm-attaining operators. We also prove that every compact operator from MFormula to {Lambda}1, v can be approximated by finite-rank norm-attaining operators.

The first author was supported by MEC project MTM2006–04837 and Junta de Andalucía ‘Proyecto de Excelencia FQM–01438’.

2000 Mathematics Subject Classification 46B20 (primary); 46E30 (secondary).


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