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 $\Vert X{\Vert }_A^2=\mathrm{tr}(A{X}^{\ast }X)$ for all $X\in {\mathbb{C}}^{n\times n}$. We determine the smallest constant $C_A$ for which $$\Vert XY-YX{\Vert }_{\mathrm{F}}^2\le C_A\Vert X{\Vert }_A^2\Vert Y{\Vert }_A^2$$ holds for all rank-one $X,Y\in {\mathbb{C}}^{n\times n}$, where $\Vert \cdot {\Vert }_{\mathrm{F}}$ 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.

arXiv comment15 pages