Year: 2026
Type: preprint
Abstract: For Hermitian matrices $A$ and $B$, we prove that $(A\join B)\oplus(A\meet B)$ is unitarily equivalent to $A\oplus B$, where the join and meet are taken in Olson's spectral order. Taking traces answers a question of Bourin and Lee: for positive semidefinite matrices, $\Tr(A\join B)=\Tr(A+B)$ if and only if $A\meet B=0$. Iterating the direct-sum identity gives a trace formula for a finite family of positive matrices. The trace of the spectral supremum equals the trace of the sum precisely when the ranges form an algebraic direct sum. For each fixed $1
Source: arXiv (Cornell University)
Authors: Aouichaoui, Mohamed Amine, Lee, Eun-Young
Institutions: University of Monastir, Kyungpook National University
Language: English
Subfield: Applied Mathematics
Field: Mathematics
Domain: Physical Sciences
Open Access status: green
OpenAlex ID: W7215050182