 
Summary: ALMOST INVARIANT HALFSPACES OF OPERATORS ON
BANACH SPACES
GEORGE ANDROULAKIS, ALEXEY I. POPOV, ADI TCACIUC,
AND VLADIMIR G. TROITSKY
Abstract. We introduce and study the following modified version of the Invariant
Subspace Problem: whether every operator T on an infinitedimensional Banach
space has an almost invariant halfspace, 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.
1. Introduction
Throughout the paper, X is a Banach space and by L(X) we denote the set of all
(bounded linear) operators on X. By a "subspace" of a Banach space we always mean
a "closed subspace". Given a sequence (xn) in X, we write [xn] for the closed linear
span of (xn).
Definition 1.1. A subspace Y of a Banach space X is called a halfspace if it is
both of infinite dimension and of infinite codimension in X.
Definition 1.2. If T L(X) and Y is a subspace of X, then Y is called almost
invariant under T, or Talmost invariant, if there exists a finite dimensional
subspace F of X such that T(Y ) Y + F.
