The Schur-Horn theorem for operators with finite spectrum

The carpenter and Schur--Horn problems for masas in finite factors

Two classical theorems in matrix theory, due to Schur and Horn, relate the eigenvalues of a self-adjoint matrix to the diagonal entries. These have recently been given a formulation in the setting of operator algebras as the Schur-Horn problem, where matrix algebras and diagonals are replaced respectively by finite factors and maximal abelian self-adjoint subalgebras (masas). There is a special case of the problem, called the carpenter problem, which can be stated as follows: for a masa A in a finite factor M with conditional expectation E_A, can each x in A with 0 <= x <= 1 be expressed as E_…

Schur-Horn theorems in II$_\infty$-factors

We describe majorization between selfadjoint operators in a $σ$-finite II$_\infty$ factor $(\mathcal{M},τ)$ in terms of simple spectral relations. For a diffuse abelian von Neumann subalgebra $\mathcal{A}\subset \mathcal{M}$ with trace-preserving conditional expectation $E_{\mathcal{A}}$, we characterize the closure in the measure topology of the image through $E_{\mathcal{A}}$ of the unitary orbit of a selfadjoint operator in $\mathcal{M}$ in terms of majorization (i.e., a Schur-Horn theorem). We also obtain similar results for the contractive orbit of positive operators in $\mathcal{M}$ and …

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

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 n…

Multivariable Schur-Horn theorems

We prove a variety of results describing the possible diagonals of tuples of commuting hermitian operators in type $II_1$ factors. These results are generalisations of the classical Schur-Horn theorem to the infinite dimensional, multivariable setting. Our description of these possible diagonals uses a natural generalisation of the classical notion of majorization to the multivariable setting. In the special case when both the given tuple and the desired diagonal have finite joint spectrum, our results are complete. When the tuples do not have finite joint spectrum, we are able to prove strong…

Thompson's theorem for II_1 factors

A theorem of Thompson provides a non-self-adjoint variant of the classical Schur-Horn theorem by characterizing the possible diagonal values of a matrix with given singular values. We prove an analogue of Thompson's theorem for II_1 factors.

Short proofs of theorems of Mirsky and Horn on diagonals and eigenvalues of matrices

A theorem of Mirsky provides necessary and sufficient conditions for the existence of an N-square complex matrix with prescribed diagonal entries and prescribed eigenvalues. We give a simple inductive proof of this theorem.

自伴算子的对角线

Diagonals of self-adjoint operators

自伴n × n矩阵A的特征值可以放入递减序列$λ=(λ_1,...,λ_n)$中，根据重数重复，A的对角线是$R^n$中与$λ$有某种关系的点。Schur-Horn定理用一组线性不等式来描述这种关系。我们给出了无穷维希尔伯特空间上正迹类算子的后一个结果的一个新证明，推广了我们之一关于投影对角线的结果。我们还建立了一个适当的对应的Schur不等式，它将自伴算子在$II_1$因子中的谱性质与它们在极大交换子代数上的条件期望下的像联系起来。

C*-代数Schur乘积定理、Pólya-Szegő-Rudin问题与Novak猜想

C*-algebraic Schur product theorem, Pólya-Szegő-Rudin question and Novak's conjecture

本文将Schur乘积推广到任意C*-代数上，推导出Schur和Vybíral的结果，提出并解决了交换有单位C*-代数上的Novak猜想，同时提出了C*-代数Schur乘积的Pólya-Szegő-Rudin问题。