bibkey: bourinlee2021inradius authors: J.-C. Bourin and E.-Y. Lee year: 2021 title: “Eigenvalue inequalities for positive block matrices with the inradius of the numerical range” doi: 10.1142/S0129167X22500094 url: https://arxiv.org/abs/2111.15180v1 claim: “Let $$\begin{gathered} X \\ in \\ mathbb{M}_n \end{gathered}$$. If the inequality \\left\\| \\begin{bmatrix} A &X \\\\ X^* &B\\end{bmatrix}\\right\\|_\\infty \\le \\|A+B\\|_\\infty holds for all positive block-matrix with $X$ as off-diagonal block, then $X$ is essentially Hermitian.” strata_touched:

• D5/S3/Quantum/BlockNorm/SpikeEdgeEstimate
• D5/S3/Quantum/BlockNorm/EssentiallyHermitian license: citation-only triage: anchor

Positive block matrices and essentially Hermitian off-diagonal blocks

Verified locator

DOI: 10.1142/S0129167X22500094

Source: https://arxiv.org/abs/2111.15180v1

Conjecture 3.3 is on printed page 6 of the arXiv v1 PDF. Crossref identifies the same title and authors with the IJM DOI above. Theorem 3.2 on the same page quotes Hayashi’s Theorem 2.5 (arXiv:1808.00181).

Conjecture 3.3, verbatim, with its displayed formula:

Let $X\in {\mathbb{M}}_n$. If the inequality

$${\Vert \left[\begin{array}{cc}A& X\\ X^{\ast }& B\end{array}\right]\Vert }_{\infty }\le \Vert A+B{\Vert }_{\infty }$$

holds for all positive block-matrix with $X$ as off-diagonal block, then $X$ is essentially Hermitian.

The preceding sentence is: “If W(T) is line segment, then T is a so-called essentially Hermitian matrix.” The binding convention in https://github.com/the-omega-institute/trureturing/issues/13752 encodes this as the literal affine Hermitian expression $X=\alpha H+\beta I$, with $H=H^{\ast }$ and $\alpha ,\beta \in \mathbb{C}$. The claim quantifies over every $n\ge 1$, all complex $n\times n$ matrices $X$, and every positive semidefinite completion. The norm is the Euclidean operator norm, identified in Lean with Matrix.toEuclideanCLM by definitional equality.

Hayashi’s quoted theorem says, verbatim:

Suppose that $X\in {\mathbb{M}}_n$ is invertible with $n$ distinct singular values. If the inequality

$${\Vert \left[\begin{array}{cc}A& X\\ X^{\ast }& B\end{array}\right]\Vert }_{\infty }\le \Vert A+B{\Vert }_{\infty }$$

holds for all positive block-matrix with $X$ as off-diagonal block, then $X$ is normal.

Proposition 3.4 on printed page 6 states the Frobenius-norm characterization: the same inequality for every positive completion holds if and only if $X$ is normal. This is a proved result in the source; the operator-norm claim has the stronger conclusion of an affine Hermitian matrix. The argument for other unitarily invariant norms is not supplied here.

Formal mechanism

Positive scalar shifts force the top and bottom spectral edges of $K_X(D)=\left[\begin{array}{cc}D& X\\ X^{\ast }& -D\end{array}\right]$ to sum to zero for every Hermitian $D$. Rank-one probes $D=tv{v}^{\ast }+S$, with $Sv=0$, have a two-sided edge expansion with error $1782L^4/t^3$ for $t\ge 96L$ and $L\ge \max (1,\Vert K_X(S)\Vert )$. The coefficient of $1/t$ forces normality; the coefficient of $1/t^2$, tested with the difference of two projected rank-one matrices, forces their equality. Unitary diagonalization then makes the spectrum collinear and reconstructs $X=\alpha H+\beta I$.

Code provenance

The private normal-operator and normal-matrix diagonalization proofs are adapted from the TauCeti contributors, commit 7c8d7117e41432d613582fc600dcb0a3be7dba30, under Apache-2.0. Copyright (c) 2026 The Tau Ceti contributors. The license text is available at https://www.apache.org/licenses/LICENSE-2.0. No TauCeti Lake dependency is used. Pinned Mathlib supplies the simultaneous eigenspace decomposition of the self-adjoint real and imaginary parts and the subordinate orthonormal basis construction.