Hahn-Banach extension operators and spaces of operators

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 $Z=Z_{\Phi \phi }$.