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BEST POSSIBLE COMPACTNESS RESULTS OF LIONS–PEETRE TYPE

Published online by Cambridge University Press:  20 January 2009

Fernando Cobos
Affiliation:
Departamento de Análisis Matemático, Facultad de Matemáticas, Universidad Complutense de Madrid, 28040 Madrid, ES (cobos@eucmax.sim.ucm.es)
Michael Cwikel
Affiliation:
Department of Mathematics, Technion—Israel Institute of Technology, Haifa 32000, IL (mcwikel@math.technion.ac.il)
Pedro Matos
Affiliation:
Departamento de Análisis Matemático, Facultad de Matemáticas, Universidad Complutense de Madrid, 28040 Madrid, ES (cobos@eucmax.sim.ucm.es) ESTG, Instituto Politécnico de Leiria, 2400 Leiria, PT (matos@estg.iplei.pt)
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Abstract

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If $T:A_{0}\rightarrow B$ boundedly and $T:A_{1}\rightarrow B$ compactly, then a result of Lions–Peetre shows that $T:A\rightarrow B$ compactly for a certain class of spaces $A$ which are intermediate with respect to $A_{0}$ and $A_{1}$. We investigate to what extent such results can hold for arbitrary intermediate spaces $A$. The ‘dual’ case of an operator $S$ such that $S:X\rightarrow Y_{0}$ boundedly and $S:X\rightarrow Y_{1}$ compactly, is also considered, as well as similar questions for other closed operator ideals.

AMS 2000 Mathematics subject classification: Primary 46B70; 47D50

Type
Research Article
Copyright
Copyright © Edinburgh Mathematical Society 2001