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

TL;DR

This paper clarifies and corrects Hastings' approach to Lin's theorem on almost commuting matrices, providing detailed explanations, asymptotic estimates, and extending the method to cases where one matrix is normal with spectrum in a 1D subset.

Contribution

It offers a fully explained, corrected version of Hastings' method for Lin's theorem, including asymptotic estimates and an extension to normal matrices with specific spectral properties.

Findings

Provides detailed corrections to Hastings' approach

Includes asymptotic estimates for the theorem

Extends the method to normal matrices with spectrum in a 1D subset

Abstract

Lin's theorem states that for all $\epsilon > 0$, there is a $\delta > 0$ such that for all $n \geq 1$ if self-adjoint contractions $A,B \in M_n(\mathbb{C})$ satisfy $\|[A,B]\|< \delta$ then there are self-adjoint contractions $A',B' \in M_n(\mathbb{C})$ with $[A',B']=0$ and $\|A-A'\|,\|B-B'\|<\epsilon$. 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 $\mathbb{C}$.