Distance to commuting unitary matrices

Abstract

A question going back to Halmos asks when two approximately commuting matrices of a certain kind are close to genuinely commuting matrices of the same kind. It was quickly realized that dimension-independent results were far more difficult to obtain, and in work of Lin the result was settled for self-adjoint matrices. Estimates for the distance to commuting self-adjoint matrices were sought out by many authors until optimal bounds were established by Kachkovskiy and Safarov. On the other hand, it has long been known, since the work of Voiculescu, that there is an obstruction for dimension-independent approximately commuting unitary matrices to be close to commuting unitary matrices. However, in work of Gong and Lin, and of Eilers, Loring and Pedersen, it was shown that when this obstruction vanishes a positive result still holds. Despite this, quantitative bounds for the distance in terms of the commutator of the unitary matrices were still unknown.

In this talk I will report on joint work with Hall and Kachkovskiy where we show that under the vanishing of said obstruction, we can find bounds for the distance to commuting unitary matrices in terms of the commutator of the original pair of unitary matrices.