Invariant subspace problem

Remarks

The problem is open. The hypotheses rule out easy cases. In a finite-dimensional complex space of dimension at least $2$, an eigenvector spans an invariant line. In a non-separable space, the closed span of $x, Tx, T^2 x, \ldots$ for any $x \ne 0$ is a nontrivial invariant subspace.

Positive answers are known for many classes of operators. Compact operators have nontrivial invariant subspaces (Aronszajn and Smith, 1954), and so do operators that commute with a nonzero compact operator, by Lomonosov's theorem. Normal operators have them by the spectral theorem, and subnormal operators have them by a theorem of Scott Brown. Bernstein and Robinson settled the case of polynomially compact operators in 1966, a question that Halmos had posed for operators whose square is compact.

For general Banach spaces the answer is negative. Per Enflo constructed a bounded operator on a separable Banach space with no nontrivial closed invariant subspace, and Charles Read gave such an operator on $\ell^1$.