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Duality of the weak essential norm


Author: Hans-Olav Tylli
Journal: Proc. Amer. Math. Soc. 129 (2001), 1437-1443
MSC (2000): Primary 47A30, 46B20, 46B28
DOI: https://doi.org/10.1090/S0002-9939-00-05937-2
Published electronically: October 24, 2000
MathSciNet review: 1814170
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Abstract:

It is established by an example that the natural quotient norms $S \mapsto\mathrm{dist}(S,W(E,F))$ and $S \mapsto\mathrm{dist}(S^{*},W(F^{*},E^{*}))$ are not comparable in general. Hence there is no uniform quantitative version of Gantmacher's duality theorem for weakly compact operators in terms of the preceding weak essential norm. Above $W(E,F)$ stands for the class of weakly compact operators $E\to F$, where $E$ and $F$ are Banach spaces. The counterexample is based on a renorming construction related to weakly compact approximation properties that is applied to the Johnson-Lindenstrauss space $JL$.


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Additional Information

Hans-Olav Tylli
Affiliation: Department of Mathematics, University of Helsinki, P. O. Box 4 (Yliopistonkatu 5), FIN-00014 University of Helsinki, Finland
Email: hojtylli@cc.helsinki.fi

DOI: https://doi.org/10.1090/S0002-9939-00-05937-2
Received by editor(s): August 17, 1999
Published electronically: October 24, 2000
Communicated by: Dale Alspach
Article copyright: © Copyright 2000 American Mathematical Society

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