Norm estimates related to self-commutators (Q1073132)

A selfadjoint matrix H with \(tr(H)=0\) can be written as a selfcommutator \(H=[T^*,T]\). The number \(\gamma (H)=\min \{\| T\|^ 2|\) \([T^*,T]=H\}\) satisfies \(\| H\| \leq \gamma (H)\leq 2\| H\|\). For the numbers \(\gamma_ n:=\sup \{\gamma (H)|\) H selfadjoint \(n\times n\)-matrix, \(tr(H)=0\), \(\| H\| \leq 1\}\) the estimates \(\gamma_{2k-1}\geq 2((k-1)/k),\) \(\gamma_{2k}\geq 2((k-1)/(k+2))\) are derived. Thus, \(\lim_{n\to \infty}\gamma_ n=2\). Finally, \(\gamma_ 5=4/3\) is shown.