The unconditional basic sequence problem

We construct a Banach space that does not contain any infinite un- conditional basic sequence and investigate further properties of this space. For example, it has no subspace that can be written as a topological direct sum of two infinite-dimensional spaces. This property implies that every operator on the space is a strictly singular perturbation of a multiple of the identity. In par- ticular, it is either strictly singular or Fredholm with index zero. This implies that the space is not isomorphic to any proper subspace. TRINITY COLLEGE, CAMBRIDGE, CB2 1TQ ENGLAND Current address: Department of Mathematics, University College London, Gower Street, Lon- don WC1E 6BT, England E-mail address: wtg l O@phx.cam.ac.uk UNIVERSIT#, PARIS VII, U.F.R. DE MATHtMATIQUES, 2 PLACE JUSSIEU, 75251 PARIS CEDEX 05, FRANCE Current address: Equipe d'Analyse et Mathematiques Appliquees, Universite de Mamne la Vallee, 93160 Noisy le Grand, France E-mail address: maurey@logiquejussieu.fr This content downloaded from 157.55.39.105 on Fri, 07 Oct 2016 04:30:48 UTC All use subject to http://about.jstor.org/terms