Strictly singular operators and the invariant subspace problem

Properties of strictly singular operators have recently become of topical i nterest because the work of Cowers and Maurey in [GM1] and [GM2] gives (amo ng many other brilliant and surprising results, such as those in [G1] and [ G2]) Banach spaces on which every continuous operator is of form lambda I S, where S is strictly singular. So if strictly singular operators had inv ariant subspaces, such spaces would have the property that all operators on them had invariant subspaces. However, in this paper we exhibit examples o f strictly singular operators without nontrivial closed invariant subspaces . So, though it may be true that operators on the spaces of Cowers and Maur ey have invariant subspaces, yet this cannot be because of a general result about strictly singular operators. The general assertion about strictly si ngular operators is false.