Hahn-Banach extension operators and spaces of operators
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- by Åsvald Lima and Eve Oja
- Proc. Amer. Math. Soc. 130 (2002), 3631-3640
- DOI: https://doi.org/10.1090/S0002-9939-02-06615-7
- Published electronically: May 14, 2002
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Abstract:
Let $X\subseteq Y$ be Banach spaces and let $\mathcal A\subseteq \mathcal B$ be closed operator ideals. Let $Z$ be a Banach space having the Radon-Nikodým property. The main results are as follows. If $\Phi :\mathcal A(Z,X)^*\to \mathcal B(Z,Y)^*$ is a Hahn-Banach extension operator, then there exists a set of Hahn-Banach extension operators $\phi _i:X^*\to Y^*$, $i\in I$, such that $Z=\sum _{i\in I}\oplus _1 Z_{\Phi \phi _i}$, where $Z_{\Phi \phi _i}=\{z\in Z\colon \Phi (x^*\otimes z)=(\phi _i x^*)\otimes z, \forall x^*\in X^*\}$. If $\mathcal A(\hat {Z},X)$ is an ideal in $\mathcal B(\hat {Z},Y)$ for all equivalently renormed versions $\hat {Z}$ of $Z$, then there exist Hahn-Banach extension operators $\Phi :\mathcal A(Z,X)^*\to \mathcal B(Z,Y)^*$ and $\phi :X^*\to Y^*$ such that $Z=Z_{\Phi \phi }$.References
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Bibliographic Information
- Åsvald Lima
- Affiliation: Department of Mathematics, Agder College, Gimlemoen 257, Serviceboks 422, 4604 Kristiansand, Norway
- Email: Asvald.Lima@hia.no
- Eve Oja
- Affiliation: Faculty of Mathematics, Tartu University, Vanemuise 46, EE-51014 Tartu, Estonia
- Email: eveoja@math.ut.ee
- Received by editor(s): July 16, 2001
- Published electronically: May 14, 2002
- Additional Notes: The second-named author wishes to acknowledge the warm hospitality provided by Åsvald Lima and his colleagues at Agder College, where a part of this work was done in October 2000. Her visit was supported by the Norwegian Academy of Science and Letters and by Estonian Science Foundation Grant 4400
- Communicated by: N. Tomczak-Jaegermann
- © Copyright 2002 American Mathematical Society
- Journal: Proc. Amer. Math. Soc. 130 (2002), 3631-3640
- MSC (2000): Primary 46B20, 46B28, 47L05
- DOI: https://doi.org/10.1090/S0002-9939-02-06615-7
- MathSciNet review: 1920043