The Schur-Horn theorem states: There exists a Hermitian matrix \(A\in {\mathcal M}_n(\mathbb C)\) with main diagonal \(x\in {\mathbb R}^n\) and spectrum (counting multiplicity) \(y\in {\mathbb R}^n\) if and only if \(x^{\downarrow}\) is majorized by \(y^{\downarrow}\), where \(x^{\downarrow}=(x^{\downarrow}_1,\dots,x^{\downarrow}_n)\) denotes the vector obtained by rearranging the entries of \(x\) in nonincreasing order, and the majorization relationship \(x^{\downarrow}\prec y^{\downarrow}\) is defined by:
\[ \sum_{i=1}^k x^{\downarrow}_i\leq \sum_{i=1}^k y^{\downarrow}_i,\;\;k=1,\dots,n-1; \qquad \;\sum_{i=1}^n x^{\downarrow}_i= \sum_{i=1}^n y^{\downarrow}_i. \]
The author discusses non-commutative Schur-Horn theorems and an extended majorization for Hermitian matrices.
Let \({\mathcal A}\) be a unital \(*\)-subalgebra of \({\mathcal M}_n({\mathbb C})={\mathbb C}^{n\times n}\) and let \(B\) be an Hermitian matrix. Let \({\mathcal U}_n(B)\) denote the unitary orbit of \(B\) in \({\mathcal M}_n({\mathbb C})\) and let \({\mathcal E}_{\mathcal A}\) denote the trace preserving conditional expectation onto \({\mathcal A}\). In a non-commutative Schur-Horn type theorem, the author gives a spectral characterization of the set
\[ {\mathcal E}_{\mathcal A}({\mathcal U}_n(B))=\left\{{\mathcal E}_{\mathcal A}(U^*BU)\;:\;U\in {\mathcal U}_n({\mathbb C}),\;\text{unitary matrix}\right\}, \]
and a similar result for the contractive orbit of a positive semi-definite matrix \(B\). These results are used to extend the notation of majorization and submajorization between self-adjoint matrices to spectral relations that come together with extended non-commutative Schur-Horn type theorems.