Seminar in Analysis and Quantum by Søren Eilers

Title: Stability of anticommuting matrices

Abstract: A question popularized by Halmos in the 1970’s asks whether a pair of self-adjoint matrices that almost commute (||xy-yx||<epsilon) is close to a pair of exactly commuting self-adjoint matrices in a way which is uniform in the matrix size. The problem was finally resolved positively in 1995 in work by Huaxin Lin. drawing fundamentally on operator algebraic methods, and a short proof of the fact provided not long after by Peter Friis and Mikael Rørdam remains the accepted argument. Adapting the proof of Friis and Rørdam, I proved a similar result with Loring and Pedersen in 1998 pertaining to ANTI-commuting matrices, starting from knowing that ||xy+yx||<epsilon. I will outline these quite intuitive proofs, and discuss the many new insights that have been brought forward since then, often driven by questions in mathematical physics.