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Hahn-Banach extension operators and spaces of operators

Authors:	Å svald Lima Eve Oja

Institution:	Department of Mathematics, Agder College, Gimlemoen 257, Serviceboks 422, 4604 Kristiansand, Norway ; Faculty of Mathematics, Tartu University, Vanemuise 46, EE-51014 Tartu, Estonia

Abstract:	Let $X\subseteq Y$ be Banach spaces and let $\mathcal A\subseteq \mathcal B$ be closed operator ideals. Let 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 , then there exist Hahn-Banach extension operators $\Phi:\mathcal A(Z,X)^\to \mathcal B(Z,Y)^$ and $\phi:X^\to Y^$ such that $Z=Z_{\Phi\phi}$ .

Keywords:	Hahn-Banach extension operators spaces of operators operator ideals Radon-Nikod\'{y}m property

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