Crossed-Heinz Inequalities: The Two-Dimensional Threshold and Higher-Dimensional Stability
Abstract
For positive matrices A,B and 0<t<1, Bourin asked whether the crossed-Heinz expression Ct(A,B)=AtB1−t+BtA1−t satisfies UINCt(A,B)≤ UINA+B for every unitarily invariant norm. The most direct route to this inequality is to compare Ct(A,B) with the ordinary Heinz expression Ht(A,B)=AtB1−t+A1−tBt, because the classical Heinz inequality gives UINHt(A,B)≤UINA+B. This route, however, encounters a genuine dimensional obstruction: the stronger comparison UINCt(A,B)≤UINHt(A,B) is known to fail for positive definite 3×3 matrices, and hence, by block-diagonal embedding, in every dimension n≥3. We first determine the exact threshold for this obstruction. For every pair of positive 2×2 matrices and every 0<t<1, we prove the weak-majorization relation s(Ct(A,B))≺w s(Ht(A,B)).
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BibTeX
@misc{sababheh2026crossedheinz,
title = {{Crossed-Heinz Inequalities: The Two-Dimensional Threshold and Higher-Dimensional Stability}},
author = {Mohammad Sababheh and Jean-Christophe Bourin},
howpublished = {arXiv},
year = {2026},
month = oct,
doi = {10.48550/arxiv.2610.04567},
eprint = {2610.04567},
archivePrefix = {arXiv},
url = {https://arxiv.org/abs/2610.04567},
}
RIS
TY - UNPB
TI - Crossed-Heinz Inequalities: The Two-Dimensional Threshold and Higher-Dimensional Stability
AU - Mohammad Sababheh
AU - Jean-Christophe Bourin
PY - 2026
DA - 2026/10/03
PB - arXiv
DO - 10.48550/arxiv.2610.04567
UR - https://arxiv.org/abs/2610.04567
AB - For positive matrices A,B and 0<t<1, Bourin asked whether the crossed-Heinz expression C_t(A,B)=A^tB^1−t+B^tA^1−t satisfies UINC_t(A,B)≤ UINA+B for every unitarily invariant norm. The most direct route to this inequality is to compare C_t(A,B) with the ordinary Heinz expression H_t(A,B)=A^tB^1−t+A^1−tB^t, because the classical Heinz inequality gives UINH_t(A,B)≤UINA+B. This route, however, encounters a genuine dimensional obstruction: the stronger comparison UINC_t(A,B)≤UINH_t(A,B) is known to fail for positive definite 3×3 matrices, and hence, by block-diagonal embedding, in every dimension n≥3. We first determine the exact threshold for this obstruction. For every pair of positive 2×2 matrices and every 0<t<1, we prove the weak-majorization relation s(C_t(A,B))≺_w s(H_t(A,B)).
ER -