Summary: ALMOST INVARIANT HALF-SPACES OF OPERATORS ON
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 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.
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 half-space 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 T-almost invariant, if there exists a finite dimensional
subspace F of X such that T(Y ) Y + F.