An Extremal Formula for Elliptical Width and the Bourin--Lee Conjecture
Abstract
Let X∈ Mn(ℂ), n≥2, and let δ2(X) = supdim S=2δ(XS), where XS denotes the compression of X to S, and δ(XS) is the diameter of the largest disk contained in the numerical range W(XS). Bourin and Lee proved that every positive block matrix M= A&X X*&B satisfies normM≤normA+B+δ2(X). In the same work, they conjectured that the inequality normM≤normA+B for every positive block matrix with prescribed off-diagonal block X characterizes essentially Hermitian matrices. We resolve this conjecture by proving the stronger exact formula sup(A&X X*&B)≥0 { normA&X X*&B −normA+B } = δ2(X). Thus the elliptical width is precisely the optimal norm defect associated with a fixed off-diagonal block. The reverse inequality is obtained by lifting an arbitrary two-dimensional compression of X to an explicit family of positive block matrices, yielding quantitative two-sided estimates with error of order t−1. As a consequence, the norm inequality above holds universally if and only if X is essentially Hermitian, proving Conjecture 3.3 of Bourin and Lee. We also construct an explicit 3×3 normal example showing that the radius constant in their normal-off-diagonal eigenvalue estimate is optimal at the leading eigenvalue level. By direct-sum amplification of this construction, we further prove that the same radius constant is sharp at every eigenvalue level j≥0, thereby resolving their corresponding sharpness question.
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BibTeX
@misc{sababheh2026extremal,
title = {{An Extremal Formula for Elliptical Width and the Bourin--Lee Conjecture}},
author = {Mohammad Sababheh},
howpublished = {arXiv},
year = {2026},
month = oct,
doi = {10.48550/arxiv.2610.04377},
eprint = {2610.04377},
archivePrefix = {arXiv},
url = {https://arxiv.org/abs/2610.04377},
}
RIS
TY - UNPB
TI - An Extremal Formula for Elliptical Width and the Bourin--Lee Conjecture
AU - Mohammad Sababheh
PY - 2026
DA - 2026/10/03
PB - arXiv
DO - 10.48550/arxiv.2610.04377
UR - https://arxiv.org/abs/2610.04377
AB - Let X∈ M_n(ℂ), n≥2, and let δ_2(X) = sup_dim S=2δ(X_S), where X_S denotes the compression of X to S, and δ(X_S) is the diameter of the largest disk contained in the numerical range W(X_S). Bourin and Lee proved that every positive block matrix M= A&X X^*&B satisfies normM≤normA+B+δ_2(X). In the same work, they conjectured that the inequality normM≤normA+B for every positive block matrix with prescribed off-diagonal block X characterizes essentially Hermitian matrices. We resolve this conjecture by proving the stronger exact formula sup_(A&X X^*&B)≥0 { normA&X X^*&B −normA+B } = δ_2(X). Thus the elliptical width is precisely the optimal norm defect associated with a fixed off-diagonal block. The reverse inequality is obtained by lifting an arbitrary two-dimensional compression of X to an explicit family of positive block matrices, yielding quantitative two-sided estimates with error of order t^−1. As a consequence, the norm inequality above holds universally if and only if X is essentially Hermitian, proving Conjecture 3.3 of Bourin and Lee. We also construct an explicit 3×3 normal example showing that the radius constant in their normal-off-diagonal eigenvalue estimate is optimal at the leading eigenvalue level. By direct-sum amplification of this construction, we further prove that the same radius constant is sharp at every eigenvalue level j≥0, thereby resolving their corresponding sharpness question.
ER -