Two weeks ago, an unassuming-looking paper titled On the invariant subspace problem in Hilbert spaces was submitted to the arXiv preprint server. The manuscript is only thirteen pages long, and its list of citations contains a single entry.
The paper claims to contain the final piece of the invariant subspace problem, a jigsaw puzzle that mathematicians have been working on for over half a century.
Famous unsolved problems are frequently tackled by ambitious, interesting individuals seeking to create a name for themselves. However, such efforts are typically swiftly dismissed by experts.
Per Enflo, a Swedish mathematician and author of this note, is not an ambitious up-and-comer. He is nearly 80 years old, has made a reputation for himself by solving unsolved problems, and has extensive experience with the issue at hand.
Mathematics, music, and a live gander, according to Enflo
Enflo, who was born in 1944 and is now an emeritus professor at Kent State University in Ohio, has had a remarkable career in mathematics and music.
He is a renowned concert pianist who has performed and recorded a multitude of piano concertos, as well as performed solo and with orchestras all over the globe.
Enflo is also one of the greatest problem-solvers in the functional analysis field. In addition to his work on the invariant subspace problem, Enflo also solved the basis problem and the approximation problem, both of which had remained unsolved for over four decades.
Enflo’s solution to the approximation problem allowed him to solve the equivalent Mazur’s gander problem. In 1936, the Polish mathematician Stanisaw Mazur offered a live goose to anyone who could solve his problem. In 1972, he kept his word and presented the fowl to Enflo.
What exactly is an invariant subspace?
Now we know who the protagonist is. But what about the problem of invariant subspace itself?
You will have encountered vectors, matrices, and eigenvectors if you’ve ever taken a university-level linear algebra course in the first year. If you haven’t already, a vector is an arrow with a length and direction that resides in a specific vector space. (There are numerous distinct vector spaces with varying dimensions and varying principles.)
A matrix can transform a vector by modifying its direction and/or length. If a matrix only modifies the length of a vector, with the direction remaining the same or inverted, we refer to the vector as an eigenvector of the matrix.
These lines are invariant under this matrix. Collectively, we refer to these lines as invariant subspaces of the matrix.
Eigenvectors and invariant subspaces have applications outside of mathematics; for instance, Google’s success has been attributed to “the $25 billion eigenvector.”
What about infinitely-dimensional spaces?
This is therefore an invariant subspace. The invariant subspace problem involves spaces with an infinite number of dimensions and questions if every linear operator (the equivalent of a matrix) in such spaces must have an invariant subspace.
The invariant subspace problem questions whether every bounded linear operator T on a complex Banach space X admits a non-trivial invariant subspace M of X, such that T(M) is contained back in M.
In this form, the invariant subspace problem was posed in the middle of the twentieth century, and all attempts to solve it failed.
As is typical when mathematicians are unable to solve a problem, we shift the goalposts. Mathematicians working on this problem restricted their attention to specific classes of spaces and operators.
Enflo made the initial breakthrough in the 1970s, although his result was not published until 1987. By constructing an operator on a Banach space lacking a nontrivial invariant subspace, he provided a negative response to the problem.
What’s new about this solution proposal?
How does the invariant subspace problem currently stand? Why has Enflo solved it again if he already did so in 1987?
Enflo, however, resolved the issue for Banach spaces in general. There is, however, a particularly significant type of Banach space known as a Hilbert space, which has a strong sense of geometry and is utilised extensively in physics, economics, and applied mathematics.
Enflo asserts that he has solved the invariant subspace problem for operators on Hilbert spaces, which has been notoriously challenging.
This time, Enflo responds in the affirmative, arguing in his paper that every bounded linear operator on a Hilbert space has an invariant subspace.
Expert evaluation is not yet complete
I have not gone line by line through Enflo’s preprint. According to reports, Enflo is sceptical of the proposed solution because it has not yet been evaluated by experts.
Enflo’s earlier proof for Banach spaces in general required several years of peer review. However, that paper contained more than 100 pages, so a review of the new paper’s 13 pages should be considerably quicker.
If accurate, it will be a remarkable accomplishment, particularly for a person who has already generated so many remarkable accomplishments over such a long period of time. Enflo’s numerous mathematical contributions and solutions to numerous unsolved problems have had a significant impact on the field, spawning new techniques and concepts.
I’m excited to learn whether Enflo’s work finally solves the invariant subspace problem and to observe any new mathematics that may result from this conclusion.
