J-自伴矩阵的惯性敏感Kreiss界

中文摘要 AI 辅助

设 $A \in \mathbb{C}^{n \times n}$ 的谱包含在闭单位圆盘内。其最大幂增长 $\operatorname{Power}(A):= \sup_{k \ge 0} \Vert{}A^k\Vert{}$ 用于衡量瞬态放大，而 Kreiss 常数 $\operatorname{Kreiss}(A):= \sup_{\vert{}z\vert{}>1} (\vert{}z\vert{}-1) \Vert{}(zI-A)^{-1}\Vert{}$ 用于衡量该圆盘外对应的预解式增长。经典有限维 Kreiss 定理给出 $\operatorname{Power}(A) \le en \operatorname{Kreiss}(A)$，且对于一般矩阵而言，对 $n$ 的线性依赖是不可避免的。我们证明，对于关于不定度量自伴的矩阵，环境维数 $n$ 可由极小多项式和度量惯性确定的有效维数替代。具体而言，若 $A^*J = JA$，其中 $J$ 是具有惯性 $(n-q,q)$ 的基本对称算子，则 $\operatorname{Power}(A) \le e \min\{d(A), 2q+1, 2(n-q)+1\} \operatorname{Kreiss}(A)$，这里 $d(A)$ 是极小多项式的次数。我们的证明不需要对可对角化性或谱的实性作出任何假设，而是将每个循环轨道与有限秩自伴 Hankel 算子相关联，并将其秩和惯性转移到系数估计中。基于缩放幂零移位的例子表明，即使当该惯性指标相对于矩阵大小可忽略且 $d(A)=n$ 时，对较小惯性指标的线性依赖也是渐近尖锐的。我们还得到了缩放圆盘衰减估计以及对任意非奇异 Hermitian 度量的加权范数扩展。

英文摘要

Let $A \in \mathbb{C}^{n \times n}$ have spectrum in the closed unit disk. Its maximal power growth $\operatorname{Power}(A) := \sup_{k \ge 0} \Vert{}A^k\Vert{}$ measures transient amplification, whereas the Kreiss constant $\operatorname{Kreiss}(A) := \sup_{\vert{}z\vert{}>1} (\vert{}z\vert{}-1) \Vert{}(zI-A)^{-1}\Vert{}$ measures the corresponding resolvent growth outside the disk. The classical finite-dimensional Kreiss theorem gives $\operatorname{Power}(A) \le en \operatorname{Kreiss}(A)$, and the linear dependence on $n$ is unavoidable for general matrices. We show that, for matrices selfadjoint with respect to an indefinite metric, the ambient dimension $n$ can be replaced by an effective dimension determined by the minimal polynomial and the inertia of the metric. Specifically, if $A^*J = JA$, where $J$ is a fundamental symmetry with inertia $(n-q,q)$, then $\operatorname{Power}(A) \le e \min\{d(A), 2q+1, 2(n-q)+1\} \operatorname{Kreiss}(A)$, where $d(A)$ is the degree of the minimal polynomial. Our proof requires no assumption on diagonalizability or reality on the spectrum. Instead, we associate each cyclic orbit with a finite-rank selfadjoint Hankel operator and transfer its rank and inertia to a coefficient estimate. Examples based on scaled nilpotent shifts show that the linear dependence on the smaller inertia index is asymptotically sharp, even when this index is negligible relative to the matrix size and $d(A)=n$. We also obtain scaled-disk decay estimates and weighted-norm extensions to arbitrary nonsingular Hermitian metrics.

发表机构

• VinUniversity（越南国立大学）

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