Invariant subspace problem for $\ell^2(\mathbb{N})$
Solution 1:
The answer is, we don't know.
The invariant subspace problem is known for non separable Hilbert spaces and for finite dimensional ones, e.g. see this answer here. As noted by Disintegrating By Parts, every infinite dimensional and separable Hilbert space is isometrically isomorphic to $\ell^2(\mathbb{N})$.
Hence, the question is equivalent to asking whether every bounded linear operator $T \colon \mathcal{H} \to \mathcal{H}$ has an non-trivial invariant subspace, where $\mathcal{H}$ is a separable Hilbert space.