Update 05/02/2013: Cowen and Gallardo say that a problem has been found in their proof and they no longer claim an answer to the invariant subspace problem.
At the congress of la Real Sociedad Matemática Española yesterday, Eva Gallarda and Carl Cowen presented an affirmative answer to the invariant subspace problem on separable Hilbert spaces. While it isn’t a Millennium Prize problem, it’s one of the big open problems in maths. As far as I can tell, it hasn’t been through any formal peer review yet, but they’re serious people and you’ve got to be quite sure about this kind of thing before announcing it at such a high-profile event.
The invariant subspace problem is the question of whether every bounded linear operator $T: H \rightarrow H$ has a non-trivial closed $T$-invariant subspace. The answer has been known to be “yes” on some classes of spaces and “no” on others, but an answer for separable Hilbert spaces was the most keenly sought and, until now, unknown. According to our resident analyst David Cushing, we already know that the conjecture is true in the cases of finite-dimensional and non-separable Hilbert spaces, so there was only the case of spaces isomorphic to $\ell^2$ left to check to have an answer for all Hilbert spaces.
The answer “yes” means that there is no bounded linear operator $T$ on a Hilbert space $H$ such that for every non-zero vector $x$, the vector space generated by the sequence $\{T^n(x): n \geq 0\}$ is norm dense in $H$.
We’ll keep you updated as more information appears.
Update 27/01/2013: A few people have asked what consequences an answer to the invariant subspace problem has, or why it’s worth investigating. There’s a question on MathOverflow titled “Is the Invariant Subspace Problem interesting?“, to which Bill Johnson has written a good answer.
More information
Carl Cowen y Eva Gallardo presentan la solución afirmativa al “problema del subespacio invariante”, la Real Sociedad Matemática Española (Google translation)
The invariant subspace problem – decent explanation of the problem in the Nieuw Archief voor Wiskunde, in English.
This post seems to be the only piece of originally-English-language reporting that I’ve seen on Cowen and Galliardo’s announcement… I look forward to any future updates.
BTW (as I commented on gplus) the article you link to as an exposition of the ISP does a good job with the maths, but I have reservations about the accuracy of its history; the fact that the author does not acknowledge his reliance on sections of Beauzamy’s book is poor *historical writing*.