发表机构
Abdullah Al Salem University; Princess Sumaya University for Technology（阿卜杜拉·萨勒姆大学; 公主苏玛技术大学）
机构由 AI 辅助整理，请以论文原文为准。AI 中文总结
本文证明了椭圆宽度等于固定非对角块正分块矩阵范数与对角块范数之和的最优缺陷，从而解决了 Bourin--Lee 猜想，并构造例子证明其半径常数在各级特征值上最优。
AI 中文摘要
设 \\(X\in M_n(\mathbb C)\\)，\\(n\geq2\\)，并令 \\[ \delta_2(X) = \sup_{\dim S=2}\delta(X_S), \\] 其中 \\(X_S\\) 表示 \\(X\\) 到子空间 \\(S\\) 的压缩，\\(\delta(X_S)\\) 是数值域 \\(W(X_S)\\) 内所含最大圆盘的直径。Bourin 和 Lee 证明了每个正分块矩阵 \\[ M= \begin{pmatrix} A&X\\\\ X^*&B \end{pmatrix} \\] 满足 \\[ \norm{M}\leq\norm{A+B}+\delta_2(X). \\] 在同一工作中，他们猜想：对于具有给定非对角块 \\(X\\) 的每个正分块矩阵，不等式 \\[ \norm{M}\leq\norm{A+B} \\] 成立当且仅当 \\(X\\) 本质 Hermitian。我们通过证明更强的精确公式解决了这个猜想： \\[ \sup_{\left(\begin{smallmatrix}A&X\X^*&B\end{smallmatrix}\right)\geq0} \left\{ \norm{\begin{pmatrix}A&X\X^*&B\end{pmatrix}} -\norm{A+B} \right\} = \delta_2(X). \\] 因此，椭圆宽度恰好是与固定非对角块相关的范数最优缺陷。反向不等式通过将 \\(X\\) 的任意二维压缩提升到显式的正分块矩阵族获得，从而得到误差阶为 \\(t^{-1}\\) 的定量双侧估计。作为推论，上述范数不等式对所有正分块矩阵成立当且仅当 \\(X\\) 本质 Hermitian，从而证明了 Bourin 和 Lee 的猜想 3.3。我们还构造了一个显式的 \\(3\times3\\) 正规矩阵例子，表明在其正规非对角特征值估计中，半径常数在主特征值水平是最优的。通过该构造的直和放大，我们进一步证明了同一半径常数在每个特征值水平 \\(j\geq0\\) 都是最优的，从而解决了他们相应的最优性问题。
英文摘要
Let \(X\in M_n(\mathbb C)\), \(n\geq2\), and let \[ δ_2(X) = \sup_{\dim S=2}δ(X_S), \] 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 \[ M= \begin{pmatrix} A&X X^*&B \end{pmatrix} \] satisfies \[ \norm{M}\leq\norm{A+B}+δ_2(X). \] In the same work, they conjectured that the inequality \[ \norm{M}\leq\norm{A+B} \] 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 \[ \sup_{\left(\begin{smallmatrix}A&X\\X^*&B\end{smallmatrix}\right)\geq0} \left\{ \norm{\begin{pmatrix}A&X\\X^*&B\end{pmatrix}} -\norm{A+B} \right\} = δ_2(X). \] 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 \(3\times3\) 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 \(j\geq0\), thereby resolving their corresponding sharpness question.
