Editing Invariant subspace (section)

==Almost-invariant halfspaces==

Related to invariant subspaces are so-called almost-invariant-halfspaces ('''AIHS's''').  A closed subspace <math>Y</math> of a Banach space <math>X</math> is said to be '''almost-invariant''' under an operator <math>T \in \mathcal{B}(X)</math> if <math>TY \subseteq Y+E</math> for some finite-dimensional subspace <math>E</math>; equivalently, <math>Y</math> is almost-invariant under <math>T</math> if there is a [[finite-rank operator]] <math>F \in \mathcal{B}(X)</math> such that <math>(T+F)Y \subseteq Y</math>, i.e. if <math>Y</math> is invariant (in the usual sense) under <math>T+F</math>. In this case, the minimum possible dimension of <math>E</math> (or rank of <math>F</math>) is called the '''defect'''.

Clearly, every finite-dimensional and finite-codimensional subspace is almost-invariant under every operator.  Thus, to make things non-trivial, we say that <math>Y</math> is a halfspace whenever it is a closed subspace with infinite dimension and infinite codimension.

The AIHS problem asks whether every operator admits an AIHS.  In the complex setting it has already been solved; that is, if <math>X</math> is a complex infinite-dimensional Banach space and <math>T \in \mathcal{B}(X)</math> then <math>T</math> admits an AIHS of defect at most 1.  It is not currently known whether the same holds if <math>X</math> is a real Banach space.  However, some partial results have been established: for instance, any [[self-adjoint operator]] on an infinite-dimensional real Hilbert space admits an AIHS, as does any strictly singular (or compact) operator acting on a real infinite-dimensional reflexive space.