The structure of strong linear preservers of gw-majorization on $ \mathbf{M}_{{n,m}

Linear preservers of majorization on $\ell^p(I)$

In this paper, using doubly stochastic operators, we have extended the notion of majorization to the space $\ell^p(I)$, where $I$ is assumed to be an infinite set, and then, in the case $p\in (1,+\infty)$, characterize the structure of all bounded linear maps on this space which preserves majorization.

Factorization, majorization, and domination for linear relations

Let ${\mathfrak H}_A$, ${\mathfrak H}_B$, and ${\mathfrak H}$ be Hilbert spaces. Let $A$ be a linear relation from ${\mathfrak H}$ to ${\mathfrak H}_A$ and let $B$ be a linear relation from ${\mathfrak H}$ to ${\mathfrak H}_B$. If there exists an operator $Z \in \mathbf{B}({\mathfrak H}_B,{\mathfrak H}_A)$ such that $ZB \subset A$, then $B$ is said to dominate $A$. This notion plays a major role in the theory of Lebesgue type decompositions of linear relations and operators. There is a strong connection to the majorization and factorization in the well-known lemma of Douglas, when put in the c…

The Schur-Horn theorem for operators and frames with prescribed norms and frame operator

Let $\mathcal H$ be a Hilbert space. Given a bounded positive definite operator $S$ on $\mathcal H$, and a bounded sequence $\mathbf{c} = \{c_k \}_{k \in \mathbb N}$ of non negative real numbers, the pair $(S, \mathbf{c})$ is frame admissible, if there exists a frame $\{f_k \}_{k \in \mathbb{N}} $ on $\mathcal H$ with frame operator $S$, such that $\|f_k \|^2 = c_k$, $k \in \mathbb {N}$. We relate the existence of such frames with the Schur-Horn theorem of majorization, and give a reformulation of the extended version of Schur-Horn theorem, due to A. Neumann. We use it to get necessary conditi…

关于双曲（多项式）系统上的 $e$-双随机矩阵

On $e$-doubly stochastic matrices over hyperbolic (polynomial) systems

本文刻画了双曲系统上$e$-双随机矩阵的线性保持算子，揭示了其与优超的联系，并证明了由Jordan框架诱导的矩阵是极端点。

算子 Schmidt 秩与 Schmidt 轮廓的几何性质

Operator Schmidt Rank and the Geometry of Schmidt Profiles

本文证明了酉算子作用下 Schmidt 秩可达性的精确下界，并通过优序关系控制了整个 Schmidt 轮廓，进而得到 Schatten 范数与 Rényi 熵的尖锐界。