I was a bit surprised today to see a preprint by Per Enflo on the arXiv (arXiv:2305.15442). He is already 79 years old (Wikipedia)!

Per Enflo is famous for several fundamental results in functional analysis (see the English Wikipedia article linked above for an overview). The one related to this post is his counter-example (the first one to be found) to the invariant subspace problem in Banach spaces: He constructed a Banach space together with a bounded linear operator on it such that this operator does not have any non-trivial invariant closed subspace.

Enflo came up with this counter-example in 1975 and since then, of course, more have been found, even on the Banach space \(\ell^1\). But up to now no counter-example is known on a reflexive Banach space; and especially, not on a separable Hilbert space (on non-separable Hilbert spaces every bounded linear operator has a non-trivial invariant closed subspace for trivial reasons).

Enflo’s arXiv preprint from today proposes a proof that actually every bounded linear operator acting on a separable Hilbert space has a non-trivial invariant closed subspace, i.e. a positive solution to the invariant subspace problem in separable Hilbert spaces! If his arguments turn out to be correct, it will be quite a big deal!