PreprintCC BY-NC-ND 4.0arXiv
A sharp weighted commutator inequality for rank-one matrices
Abstract
Let A∈ℂn× n be positive definite Hermitian, where n≥ 2, and define |X|A2=tr(AX*X) for all X∈ℂn× n. We determine the smallest constant CA for which |XY−YX|F2 ≤ CA|X|A2|Y|A2 holds for all rank-one X,Y∈ℂn× n, where |·|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.
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BibTeX
@misc{pang2026sharp,
title = {{A sharp weighted commutator inequality for rank-one matrices}},
author = {Zhekai Pang},
howpublished = {arXiv},
year = {2026},
month = oct,
doi = {10.48550/arxiv.2610.29240},
eprint = {2610.29240},
archivePrefix = {arXiv},
url = {https://arxiv.org/abs/2610.29240},
note = {CC BY-NC-ND 4.0},
}
RIS
TY - UNPB
TI - A sharp weighted commutator inequality for rank-one matrices
AU - Zhekai Pang
PY - 2026
DA - 2026/10/21
PB - arXiv
DO - 10.48550/arxiv.2610.29240
UR - https://arxiv.org/abs/2610.29240
AB - Let A∈ℂ^n× n be positive definite Hermitian, where n≥ 2, and define |X|_A^2=tr(AX^*X) for all X∈ℂ^n× n. We determine the smallest constant C_A for which |XY−YX|_F^2 ≤ C_A|X|_A^2|Y|_A^2 holds for all rank-one X,Y∈ℂ^n× n, where |·|_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.
ER -