Non-commutative Schur-Horn theorems and extended majorization for hermitian matrices

TL;DR

This paper characterizes the spectral properties of unitary and contractive orbits of hermitian matrices within subalgebras, extending majorization concepts through non-commutative Schur-Horn theorems.

Contribution

It provides a spectral characterization of conditional expectations of unitary orbits and extends majorization notions to non-commutative settings.

Findings

Spectral description of conditional expectations of unitary orbits.

Extension of majorization to non-commutative matrices.

New Schur-Horn type theorems for spectral relations.

Abstract

Let $\mathcal A\subseteq \mat$ be a unital $*$-subalgebra of the algebra $\mat$ of all $n\times n$ complex matrices and let $B$ be an hermitian matrix. Let $\U_n(B)$ denote the unitary orbit of $B$ in $\mat$ and let $\mathcal E_\mathcal A$ denote the trace preserving conditional expectation onto $\mathcal A$. We give an spectral characterization of the set $$ \mathcal E_\mathcal A(\U_n(B))=\{\mathcal E_\mathcal A(U^* B U): U\in \mat,\ \text{unitary matrix}\}.$$ We obtain a similar result for the contractive orbit of a positive semi-definite matrix $B$. We then use these results to extend the notions of majorization and submajorization between self-adjoint matrices to spectral relations that come together with extended (non-commutative) Schur-Horn type theorems.