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Duality of the weak essential norm

Author(s): Hans-Olav Tylli
Journal: Proc. Amer. Math. Soc. 129 (2001), 1437-1443.
MSC (2000): Primary 47A30, 46B20, 46B28
Posted: October 24, 2000
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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: 10.1090/S0002-9939-00-05937-2
PII: S 0002-9939(00)05937-2
Received by editor(s): August 17, 1999
Posted: October 24, 2000
Communicated by: Dale Alspach
Copyright of article: Copyright 2000, American Mathematical Society

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