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 \[ C_t(A,B)=A^tB^{1-t}+B^tA^{1-t} \] satisfies \[ \UIN{C_t(A,B)}\le \UIN{A+B} \] for every unitarily invariant norm. The most direct route to this inequality is to compare $C_t(A,B)$ with the ordinary Heinz expression \[ H_t(A,B)=A^tB^{1-t}+A^{1-t}B^t, \] because the classical Heinz inequality gives $\UIN{H_t(A,B)}\le\UIN{A+B}$. This route, however, encounters a genuine dimensional obstruction: the stronger comparison $\UIN{C_t(A,B)}\le\UIN{H_t(A,B)}$ is known to fail for positive definite $3\times 3$ matrices, and hence, by block-diagonal embedding, in every dimension $n\ge 3$. We first determine the exact threshold for this obstruction. For every pair of positive $2\times 2$ matrices and every $0<t<1$, we prove the weak-majorization relation \[ s(C_t(A,B))\prec_w s(H_t(A,B)). \]