Impressions of convexity - An illustration for commutator bounds

TL;DR

This paper establishes the optimal constant for a Schatten norm inequality involving matrix commutators, using complex interpolation theory to advance understanding of matrix norm bounds.

Contribution

It determines the sharpest constant for a Schatten norm commutator inequality, providing a precise bound that was previously unknown.

Findings

Identified the exact constant $C_{p,q,r}$ for the inequality.

Applied complex interpolation theory as the main analytical tool.

Enhanced understanding of matrix commutator bounds in Schatten norms.

Abstract

We determine the sharpest constant $C_{p,q,r}$ such that for all complex matrices $X$ and $Y$, and for Schatten $p$-, $q$- and $r$-norms the inequality $$ \|XY-YX\|_p\leq C_{p,q,r}\|X\|_q\|Y\|_r $$ is valid. The main theoretical tool in our investigations is complex interpolation theory.