An algebraic characterization of non-singular matrix semicircles

Abstract

Let $A_1, \ldots, A_r$ be Hermitian $n \times n$ matrices and $S = \sum A_i \otimes s_i$ the associated matrix semicircle, where $s_1, \ldots, s_r$ are free semicircular variables. We prove that the following are equivalent: (i) the matrix pencil $A = \sum A_i x_i$ is LR-semisimple (decomposes, up to left--right equivalence, as a direct sum of unsplittable pencils); (ii) $S$ is non-singular at $t = 0$ (the matrix-valued Cauchy transform has a continuous boundary limit near the origin); (iii) the covariance map $\eta\colon X \mapsto \sum A_i X A_i$ is symmetrically DS-scalable (there exists $C \succ 0$ with $\eta(C) = C^{-1}$). When these hold, the spectral density satisfies $f(0) = \frac{1}{\pi}\,\mathrm{tr}(C)$, where $C$ is the unique trace minimizer of the solution set $\{W \succ 0 : \eta(W)\,W = I\}$. The proof combines algebraic and analytic ingredients. On the algebraic side, we establish the equivalence (i) $\Leftrightarrow$ (iii) using Gurvits' capacity theory for indecomposable maps and a geodesic reflection theorem in the Riemannian manifold of positive definite matrices, which upgrades DS-scalability to symmetric DS-scalability for self-adjoint completely positive maps. On the analytic side, we prove (iii) $\Rightarrow$ (ii) via a Lyapunov--Schmidt reduction of Speicher's equation at a trace-minimizing solution, showing that the Jacobian of the bifurcation equations is positive definite. This removes a stability hypothesis that was required in earlier approaches.