An Extremal Formula for Elliptical Width and the Bourin--Lee Conjecture

Let $X\in M_n(\mathbb{C})$, $n\ge 2$, and let $${\delta }_2(X)=\underset{\dim S=2}{\sup }\delta (X_S),$$ where $X_S$ denotes the compression of $X$ to $S$, and $\delta (X_S)$ is the diameter of the largest disk contained in the numerical range $W(X_S)$. Bourin and Lee proved that every positive block matrix $$M=\left(\begin{array}{ccc}A& X{X}^{\ast }& B\end{array}\right)$$ satisfies $$\text{norm}M\le \text{norm}A+B+{\delta }_2(X)\mathrm{.}$$ In the same work, they conjectured that the inequality $$\text{norm}M\le \text{norm}A+B$$ 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 $$\underset{\left(\begin{array}{cc}A& X\\ X^{\ast }& B\end{array}\right)\ge 0}{\sup }\{\text{norm}\left(\begin{array}{cc}A& X\\ X^{\ast }& B\end{array}\right)-\text{norm}A+B\}={\delta }_2(X)\mathrm{.}$$ 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 $3\times 3$ 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 $j\ge 0$, thereby resolving their corresponding sharpness question.