On a Norm Compression Inequality for 2XN Partitioned Block Matrices

正定分块矩阵的对角块与非对角块

Diagonal and off-diagonal blocks of positive definite partitioned matrices

该研究建立了正定分块矩阵块间的精确不等式，并提出关于三个收缩算子的三角型不等式猜想，认为不等式中常数c=3/4是精确的。

A Norm Compression Inequality for Block Partitioned Positive Semidefinite Matrices

Let $A$ be a positive semidefinite matrix, block partitioned as $$ A=\twomat{B}{C}{C^*}{D}, $$ where $B$ and $D$ are square blocks. We prove the following inequalities for the Schatten $q$-norm $||.||_q$, which are sharp when the blocks are of size at least $2\times2$: $$ ||A||_q^q \le (2^q-2) ||C||_q^q + ||B||_q^q+||D||_q^q, \quad 1\le q\le 2, $$ and $$ ||A||_q^q \ge (2^q-2) ||C||_q^q + ||B||_q^q+||D||_q^q, \quad 2\le q. $$ These bounds can be extended to symmetric partitionings into larger numbers of blocks, at the expense of no longer being sharp: $$ ||A||_q^q \le \sum_{i} ||A_{ii}||_q^q + …

A sharpened Schwarz-Pick operatorial inequality for nilpotent operators

Let denote by $S(ϕ)$ the extremal operator defined by the compression of the unilateral shift $S$ to the model subspace $ H(ϕ)=H^{2} \ominus ϕH^{2} $ as the following $S(ϕ)f(z)=P(zf(z)),$ where $P$ denotes the orthogonal projection from the Hardy space $H^{2}$ onto $ H(ϕ)$ and $ϕ$ is an inner function on the unit disc. In this mathematical notes, we give an explicit formula of the numerical radius of the truncated shift $S(ϕ)$ in the particular case where $ϕ$ is a finite Blaschke product with unique zero and an estimate on the general case. We establish also a sharpened Schwarz-Pick operatoria…

椭圆宽度的一个极值公式与Bourin--Lee猜想

An Extremal Formula for Elliptical Width and the Bourin--Lee Conjecture

本文证明了椭圆宽度等于固定非对角块正分块矩阵范数与对角块范数之和的最优缺陷，从而解决了 Bourin--Lee 猜想，并构造例子证明其半径常数在各级特征值上最优。