Invariant and Almost-Invariant Subspaces for Pairs of Idempotents

It is known that every bounded operator on an infinite dimensional separable Hilbert space ({mathcal{H}}) has an invariant subspace if and only if each pair of idempotents on ({mathcal{H}}) has a common invariant subspace. We show that the same equivalence holds for operators and pairs of idempotents that are essentially selfadjoint. We also show that each pair of idempotents on ({mathcal{H}}) has a common almost-invariant half-space.