离散群上伪谱的定位
Localisation of pseudospectra on discrete groups
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中文总结 AI 辅助
本文将此前谱相关方法推广到可数阿贝尔群上的带算子，通过局部块伪谱并集误差控制覆盖原算子伪谱，方法可推广至$\text{Z}^d$及其他可数阿贝尔群。
中文摘要 AI 辅助
本文将我们此前论文《关于谱包含集及有界线性算子谱与伪谱的计算》[J. Spectr. Theory 14 (2024), 719--804]中的两种方法及对应结果，从$\boldsymbol{\text{ℓ}^2(\boldsymbol{\text{Z}})}$上的三对角算子推广到具有可数阿贝尔群$\boldsymbol{G}$和希尔伯特空间$\boldsymbol{Y}$的$\boldsymbol{\text{ℓ}^2(G,Y)}$上的带算子$\boldsymbol{A}$。我们再次通过$A$的有限且规模适中的“局部块”伪谱的并集，以误差控制覆盖$A$的伪谱。虽然主要应用是理解诸多物理问题中隐含的$\boldsymbol{G=\text{Z}^d}$情形，但我们对所谓$\boldsymbol{\tau}$和$\boldsymbol{\tau_1}$方法的新方法可直接推广到可数阿贝尔群$\boldsymbol{G}$。
英文摘要
In this paper we generalise two of the methods and corresponding results from our previous paper ``On spectral inclusion sets and computing the spectra and pseudospectra of bounded linear operators'' [J. Spectr. Theory 14 (2024), 719--804] from tridiagonal operators on $\ell^2(\Z)$ to band operators $A$ on $\ell^2(G,Y)$ with a countable Abelian group $G$ and a Hilbert space $Y$. Again, we cover the pseudospectra of $A$, with error-control, via a union of pseudospectra of finite and moderately sized ``local patches'' of $A$. While a major application is to understand the case $G=\Z^d$ that is immanent in many physical problems, our new approach to the so-called $τ$ and $τ_1$ methods immediately extends to countable Abelian groups $G$.
