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.

We show that any bounded operator T on a separable, reflexive, infinitedimensional Banach space X admits a rank one perturbation which has an invariant subspace of infinite dimension and codimension. In the non-reflexive spaces, we show that the same is true for operators which have non-eigenvalues in the boundary of their spectrum. In the Hilbert space, our methods produce perturbations that are also small in norm, improving on an old result of Brown and Pearcy.

A closed subspace of a Banach space X is almost-invariant for a collection S of bounded linear operators on X if for each T ∈ S there exists a finite-dimensional subspace F T of X such that TY ⊆ Y + F T . In this paper, we study the existence of almostinvariant subspaces of infinite dimension and codimension for various classes of Banach and Hilbert space operators. We also examine the structure of operators which admit a maximal commuting family of almost-invariant subspaces. In particular, we prove that if T is an operator on a separable Hilbert space and if TP − PT has finite rank for all projections P in a given maximal abelian self-adjoint algebra M then T = M + F where M ∈ M and F is of finite rank.

In this paper, we study the structure of closed algebraic ideals in the algebra of operators acting on a Lorentz sequence space.For two closed ideals J 1 and J 2 in L(X), we will denote by J 1 ∧ J 2 the largest closed ideal J in L(X) such that J ⊆ J 1 and J ⊆ J 2 (that is, J 1 ∧ J 2 = J 1 ∩ J 2 ), and we will Date

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