On Hastings' approach to Lin's Theorem for Almost Commuting Matrices

Abstract

Lin's theorem states that for all $\epsilon >0$, there is a $\delta >0$ such that for all $n\ge 1$ if self-adjoint contractions $A,B\in M_n(\mathbb{C})$ satisfy $\Vert [A,B]\Vert <\delta$ then there are self-adjoint contractions $A^{\prime },B^{\prime }\in M_n(\mathbb{C})$ with $[A^{\prime },B^{\prime }]=0$ and $\Vert A-A^{\prime }\Vert ,\Vert B-B^{\prime }\Vert <\epsilon$. We present fully explained and corrected details of the approach in arXiv:0808.2474, which was the first version of Lin's theorem to provide asymptotic estimates. We also apply this method to the case where $B$ is a normal matrix with spectrum lying in some nice 1-dimensional subset of $\mathbb{C}$.

arXiv comment1st submission: 66 pages with 7 figures; 2nd sub: 57 pages with 7 figures, edited for conciseness and readability, typos and minor errors corrected (see Introduction for more information)