A sharp weighted commutator inequality for rank-one matrices

Abstract

Let $A\in\mathbb{C}^{n\times n}$ be positive definite Hermitian, where $n\ge 2$, and define $\|X\|_A^2=\mathrm{tr}(AX^*X)$ for all $X\in\mathbb{C}^{n\times n}$. We determine the smallest constant $C_A$ for which $$ \|XY-YX\|_{\mathrm{F}}^2 \le C_A\|X\|_A^2\|Y\|_A^2 $$ holds for all rank-one $X,Y\in\mathbb{C}^{n\times n}$, where $\|\cdot\|_{\mathrm{F}}$ denotes the Frobenius norm. The constant depends only on the two smallest eigenvalues of $A$ and is determined by a strictly monotone scalar equation. This answers the sharp-constant question for rank-one matrices formulated by Fang and Cheng. We also characterize all equality pairs.