## Duality of the weak essential norm

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- by Hans-Olav Tylli PDF
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**129**(2001), 1437-1443 Request permission

## 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 $JL$.## References

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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
- Received by editor(s): August 17, 1999
- Published electronically: October 24, 2000
- Communicated by: Dale Alspach
- © Copyright 2000 American Mathematical Society
- 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
- MathSciNet review: 1814170