Invariant Subspace Problem

The invariant subspace problem for Hilbert spaces asks whether every bounded operator on an infinite-dimensional separable complex Hilbert space has a nonzero proper closed invariant subspace. The subspace is required to be invariant under that operator, rather than under its full commutant. A hyperinvariant subspace would supply such a subspace, but failure to find one does not exclude ordinary invariant subspaces. The invariant-projection problem inside a specified von Neumann algebra also restricts the possible orthogonal projections, so it is a different formulation.