On Hahn-Banach extensions for certain operator ideals

Abstract

Introduction. Given an operator ideal 9.1, we say that a Banach space X has the N-extension property (N-EP) if, for every pair of Banach spaces Y and Z with Y a subspace of Z, every S ~ N (Y, X) admits an extension S ~ 9.1 (Z, X). Every such space X will be called an extension space for N. If ~3 is a further ideal and S e N(Y, X) is extended to S s ~3 (Z, X), with Y and Z as above, we say X has the (9.1, ~)-EP. In this paper, we will study the N-EP for certain closed operator ideals 9.1. A sufficient condition is obtained to guarantee that the spaces with N-EP are precisely the A~ with the Schur property. It is well-known that the ideal of all weakly compact operators belongs to the class of these ideals, but this class is in fact much larger. On the other hand, we shall see that certain ideals, which are in a sense close to the ideal of weakly compact operators, do not belong to this class. In particular, using specific properties of the well-known Tsirelson space, we show that the family of extension spaces of the ideal of Banach-Saks operators is properly larger than the family of A~ with the Schur property. For a discussion of the extension properties of operator ideals, where the domain space is fixed, the reader is referred to [22]. There the corresponding lifting properties are also studied. The contents of the present paper is part of the author's thesis, which was prepared under the supervision of Prof. H. Jarchow at the University of Ztirich. The author would like to express his gratitude to him for many helpful discussions.