Has a mathematician solved the ‘invariant subspace problem’? And what does that even mean?

By Nathan Brownlowe, University of Sydney

Two weeks ago, a modest-looking paper was uploaded to the arXiv preprint server with the unassuming title “On the invariant subspace problem in Hilbert spaces”. The paper is just 13 pages long and its list of references contains only a single entry.

The paper purports to contain the final piece of a jigsaw puzzle that mathematicians have been picking away at for more than half a century: the invariant subspace problem.

Famous open problems often attract ambitious attempts at solutions by interesting characters out to make their name. But such efforts are usually quickly shot down by experts.

However, the author of this short note, Swedish mathematician Per Enflo, is no ambitious up-and-comer. He is almost 80, has made a name for himself solving open problems, and has quite a history with the problem at hand.

Per Enflo: mathematics, music, and a live goose

Born in 1944 and now an emeritus professor at Kent State University, Ohio, Enflo has had a remarkable career, not only in mathematics but also in music.

He is a renowned concert pianist who has performed and recorded numerous piano concertos, and has performed solo and with orchestras across the world.

Enflo is also one of the great problem-solvers in a field called functional analysis. Aside from his work on the invariant subspace problem, Enflo solved two other major problems – the basis problem and the approximation problem – both of which had remained open for more than 40 years.

By solving the approximation problem, Enflo cracked an equivalent puzzle called Mazur’s goose problem. Polish mathematician Stanisław Mazur had in 1936 promised a live goose to anyone who solved his problem – and in 1972 he kept his word, presenting the goose to Enflo.

What’s an invariant subspace?

Now we know the main character. But what about the invariant subspace problem itself?

If you’ve ever taken a first-year university course in linear algebra, you will have come across things called vectors, matrices and eigenvectors.

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