We construct a strictly singular non-compact operator on Gowers' and Maurey's space GM.

Abstract. We study some questions concerning the structure of the set of spreading models of a separable infinite-dimensional Banach space X. In particular we give an example of a reflexive X so that all spreading models of X contain ℓ 1 but none of them is isomorphic to ℓ 1 . We also prove that for any countable set C of spreading models generated by weakly null sequences there is a spreading model generated by a weakly null sequence which dominates each element of C. In certain cases this ensures that X admits, for each α < ω 1 , a spreading model (x (α)i ) i is dominated by (and not equivalent to) (x (β) i ) i . Some applications of these ideas are used to give sufficient conditions on a Banach space for the existence of a subspace and an operator defined on the subspace, which is not a compact perturbation of a multiple of the inclusion map.

We introduce and study the following modified version of the Invariant Subspace Problem: whether every operator T on an infinite-dimensional Banach space has an almost invariant half-space, that is, a subspace Y of infinite dimension and infinite codimension such that Y is of finite codimension in T (Y ). We solve this problem in the affirmative for a large class of operators which includes quasinilpotent weighted shift operators on p (1 p < ∞) or c0. Mathematics Subject Classification (2000). Primary 47A15.

Any regular mixed Tsirelson space T (θ n , S n ) I N for which θn θ n → 0, where θ = lim n θ 1/n n , is shown to be arbitrarily distortable. Certain asymptotic ℓ 1 constants for those and other mixed Tsirelson spaces are calculated. Also a combinatorial result on the Schreier families (S α ) α<ω 1 is proved and an application is given to show that for every Banach space X with a basis (e i ), the two ∆-spectrums ∆(X) and ∆(X, (e i )) coincide. *

221 222 G. ANDROULAKIS ET AL. Isr. J. Math. ABSTRACT V. D. Milman proved in [20] that the product of two strictly singular operators on Lp[0, 1] (1 p < ∞) or on C[0, 1] is compact. In this note we utilize Schreier families S ξ in order to define the class of S ξ -strictly singular operators, and then we refine the technique of Milman to show that certain products of operators from this class are compact, under the assumption that the underlying Banach space has finitely many equivalence classes of Schreier-spreading sequences. Finally we define the class of S ξ -hereditarily indecomposable Banach spaces and we examine the operators on them.