Spaces of compact operators on Hilbert space with the fixed point property (Q1125284)

A Banach space \(X\) is said to contain an asymptotically isometric copy of \(c_{0}\) if there exist a null sequence \((\varepsilon_{n})\) in \((0,1)\) and a sequence \((x_{n})\) in \(X\) such that \[ \sup_{n} (1-\varepsilon_{n}) |t_{n}|\leq \biggl\|\sum_{n} t_{n}x_{n} \biggr\|\leq \sup_{n} |t_{n}| \] for all \((t_{n})\in c_{0}\). This is known to be a sufficient condition for \(X\) to fail the fixed point property; i.e., there exists a closed bounded convex subset \(C\subset X\) on which there is a nonexpansive mapping without fixed points. In this paper the authors show that every nonreflexive subspace of \(K(H)\), the space of compact operators on a Hilbert space \(H\), contains an asymptotically isometric copy of \(c_{0}\) and hence fails the fixed point property. By contrast, \textit{M. Besbes} [``Points fixes dans les espaces des opérateurs compacts'', to appear] has shown that reflexive subspaces of \(K(H)\) have the fixed point property.