Commutator Approximants

Abstract

This paper deals with minimizing $||B - (AX - XA)|{|_p}$, where $A$ and $B$ are fixed, $B \in {\mathcal {C}_p}$, and $X$ varies such that $AX - XA \in {\mathcal {C}_p}$. (Here, ${\mathcal {C}_p}$ denotes the von Neumann-Schatten class and ${\left \| \right \|_p}$ denotes its norm.) The main result (Theorem 3.2) says that if $A$ is normal and $AB = BA$ then $||B - (AX - XA)|{|_p},1 \leq p < \infty$, is minimized if and for $1 < p < \infty$ only if, $AX - XA = 0$; and that the map $X \to ||B - (AX - XA)||_p^p,1 < p < \infty$, has a critical point at $X = V$ if and only if $AV - VA = 0$.