Almost Invariant Half-spaces of Algebras of Operators

Given a Banach space X and a bounded linear operator T on X, a subspace Y of X is almost invariant under T if TY í Y+F{TY\subseteq Y+F} for some finite-dimensional “error” F. In this paper, we study subspaces that are almost invariant under every operator in an algebra \mathfrak A{\mathfrak A} of operators acting on X. We show that if \mathfrak A{\mathfrak A} is norm closed then the dimensions of “errors” corresponding to operators in \mathfrak A{\mathfrak A} must be uniformly bounded. Also, if \mathfrak A{\mathfrak A} is generated by a finite number of commuting operators and has an almost invariant half-space (that is, a subspace with both infinite dimension and infinite codimension) then \mathfrak A{\mathfrak A} has an invariant half-space.