Impressions of convexity - An illustration for commutator bounds

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 $$\Vert XY-YX{\Vert }_p\le C_{p,q,r}\Vert X{\Vert }_q\Vert Y{\Vert }_r$$ is valid. The main theoretical tool in our investigations is complex interpolation theory.

32 pages, 88 pictures