I was recently reminded of the invariant subspace problem in Hilbert spaces: the question of whether every bounded operator on a Hilbert space has a closed invariant subspace. Of famous open problems in mathematics, this one is perhaps the most surprising. It sounds like it should be exercise 7 of chapter 2 of a book on Hilbert spaces; yet the answer is still unknown. (I have no idea what the answer should be; I’m just surprised that it’s so hard to figure out one way or the other. What makes it particularly surprising is the answer is known for Banach spaces.)

B. F. Yadav has a survey article on the subject The Invariant Subspace Problem. It appears in Nieuw Archief voor Wiskunde, a publication of the Royal Dutch Mathematical Society, which prints the occasional article in English.