On Hahn-Banach extensions for certain operator ideals
Let \({\mathfrak A}\) be a closed ideal of operators between Banach spaces, and let Y and Z be Banach spaces with Y a closed subspace of Z. We exhibit an extensive class of ideals with the property that the Banach spaces X for which every \(S: Y\to X\) in \({\mathfrak A}\) may be extended to a \(\tilde S:\) \(Z\to X\) also in \({\mathfrak A}\), are precisely the \({\mathfrak L}_{\infty}\)- spaces with the Schur property. It is well known that the ideal of weakly compact operators belongs to this class. On the other hand, certain ideals, which are in a sense close to the ideal of weakly compact operators, do not belong to this class. In particular, we show that the ideal of Banach-Saks operators does not belong to this class by using specific properties of the Tsirelson space.
closed ideal of operators between Banach spaces
\({\mathfrak L}_{\infty }\)-spaces with the Schur property
scientific article; zbMATH DE number 3915051
scientific article