Strictly singular operators and the invariant subspace problem
Properties of strictly singular operators have recently become of topical interest because the work of Gowers and Maurey in [GM1] and [GM2] gives (among many other brilliant and surprising results, such as those in [G1] and [G2]) Banach spaces on which every continuous operator is of form λ I + S, where S is strictly singular. So if strictly singular operators had invariant subspaces, such spaces would have the property that all operators on them had invariant subspaces. However, in this paper we exhibit examples of strictly singular operators without nontrivial closed invariant subspaces. So, though it may be true that operators on the spaces of Gowers and Maurey have invariant subspaces, yet this cannot be because of a general result about strictly singular operators. The general assertion about strictly singular operators is false.