Invariant subspace problem
Let be a Hilbert space. The subspaces and are trivially invariant under any linear operator on , and so these are referred to as the trivial invariant subspaces. The problem is concerned with determining whether bounded operators necessarily have non-trivial invariant subspaces.
This is one of the most famous open problems in functional analysis. Enflo  constructed Banach spaces for which the corresponding question has a negative answer, and recently Argyros and Haydon constructed a Banach space for which the corresponding question has a positive answer .
For a nice overview to the problem see ,  or .
 P. Enflo, On the invariant subspace problem for Banach spaces, Acta Math. 158 (1987), 213-313. MathSciNet
 S. A. Argyros and R. G. Haydon, A hereditarily indecomposable -space that solves the scalar-plus-compact problem, arXiv:0903.3921 (2009).
* indicates original appearance(s) of problem.