广义Cesàro矩阵的每个正整数阶的超正规性
Hyponormality of Generalized Cesàro Matrices of Every Positive Integer Order
- Zhejiang University（浙江大学）
机构由 AI 辅助整理，请以论文原文为准。
AI总结：
本文通过计算机辅助证明，给出了广义Cesàro矩阵超正规的充要条件（参数非负），利用缺陷算子分解和离散Gram多项式，覆盖所有正整数阶。
AI中文摘要：
对于正整数$m$和实参数$\alpha>-1$，考虑$\ell^2(\mathbb N_0)$上的广义Cesàro矩阵，其条目为$m(n-j+1)_{m-1}/(n+\alpha+1)_m$（其中$j\le n$）。我们给出了一个计算机辅助证明，表明该算子超正规当且仅当$\alpha\ge0$。主要的代数成分是将一个辅助缺陷算子显式分解为一个正对角算子和至多$m$个秩一算子。离散Gram多项式决定了这些项的符号和系数。对于非负参数，加权估计将正性归结为标量不等式。一个统一的解析估计覆盖所有$m\ge32768$，而有限有理数证书覆盖其余阶数和整个参数区间，无需参数采样或截断无限算子。该证明还给出了自交换子的统一加权下界。对于$-1<\alpha<0$，有限支撑向量对每个阶数给出负二次型。
英文摘要：
For a positive integer $m$ and a real parameter $α>-1$, consider the generalized Cesàro matrix on $\ell^2(\mathbb N_0)$ with entries $m(n-j+1)_{m-1}/(n+α+1)_m$ for $j\le n$. We give a computer-assisted proof that this operator is hyponormal if and only if $α\ge0$. The main algebraic ingredient is an explicit decomposition of an auxiliary defect operator into a positive diagonal operator and at most $m$ rank-one operators. Discrete Gram polynomials determine the signs and coefficients of these terms. For nonnegative parameters, weighted estimates reduce positivity to scalar inequalities. A uniform analytic estimate covers every $m\ge32768$, while finite rational certificates cover the remaining orders and entire parameter intervals, without parameter sampling or truncating an infinite operator. The proof also gives a uniform weighted lower bound for the self-commutator. For $-1<α<0$, finite-support vectors give negative quadratic forms for every order.
