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Proceedings of the American Mathematical Society

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Hahn-Banach extension operators and spaces of operators

Authors: Åsvald Lima and Eve Oja
Journal: Proc. Amer. Math. Soc. 130 (2002), 3631-3640
MSC (2000): Primary 46B20, 46B28, 47L05
Published electronically: May 14, 2002
MathSciNet review: 1920043
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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 $Z=Z_{\Phi\phi}$.

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Additional Information

Åsvald Lima
Affiliation: Department of Mathematics, Agder College, Gimlemoen 257, Serviceboks 422, 4604 Kristiansand, Norway

Eve Oja
Affiliation: Faculty of Mathematics, Tartu University, Vanemuise 46, EE-51014 Tartu, Estonia

Keywords: Hahn-Banach extension operators, spaces of operators, operator ideals, Radon-Nikod\'{y}m property
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
Article copyright: © Copyright 2002 American Mathematical Society