AI 中文总结
该研究针对希尔伯特空间上的有界线性算子，定义渐近数值域与渐近数值半径，推导其相关性质，给出非平凡超不变子空间与不变子空间的存在条件，证明亚正规算子的渐近数值半径等于谱半径。
AI 中文摘要
设$\boldsymbol{\textit{H}}$为复可分希尔伯特空间，$\boldsymbol{\textit{T}}$为$\boldsymbol{\textit{H}}$上的有界线性算子，$\boldsymbol{\textit{x}}\boldsymbol{\textit{\text{∈}}}\boldsymbol{\textit{H}}$，令$\boldsymbol{\textit{W}}_{\boldsymbol{\textit{a}}}(\boldsymbol{\textit{T}},\boldsymbol{\textit{x}})$为序列$\boldsymbol{\textit{\text{⟨}}}\boldsymbol{\text{|}}\boldsymbol{\textit{T}}^{\boldsymbol{\textit{n}}}\boldsymbol{\text{|}}^{\boldsymbol{\text{1/}}\boldsymbol{\textit{n}}}\boldsymbol{\textit{x}}\boldsymbol{\text{,}}\boldsymbol{\textit{x}}\boldsymbol{\text{⟩}}$（$\boldsymbol{\textit{n}}\boldsymbol{\text{=}}\boldsymbol{\text{1}}\boldsymbol{\text{,}}\boldsymbol{\text{2}}\boldsymbol{\text{,}}\boldsymbol{\text{…}}$）的聚点集。分别定义$\boldsymbol{\textit{T}}$的渐近数值域$\boldsymbol{\textit{W}}_{\boldsymbol{\textit{a}}}(\boldsymbol{\textit{T}})\boldsymbol{\text{:=}}\bigcup_{\boldsymbol{\text{||}}\boldsymbol{\textit{x}}\boldsymbol{\text{||}}\boldsymbol{\text{=}}\boldsymbol{\text{1}}\boldsymbol{\textit{W}}_{\boldsymbol{\textit{a}}}(\boldsymbol{\textit{T}},\boldsymbol{\textit{x}})$与渐近数值半径$\boldsymbol{\textit{w}}_{\boldsymbol{\textit{a}}}(\boldsymbol{\textit{T}})\boldsymbol{\text{:=}}\boldsymbol{\text{sup}}\boldsymbol{\textit{W}}_{\boldsymbol{\textit{a}}}(\boldsymbol{\textit{T}})$。研究表明：若存在非零向量$\boldsymbol{\textit{x}}_{\boldsymbol{\text{0}}}\boldsymbol{\text{∈}}\boldsymbol{\textit{H}}$满足$\boldsymbol{\textit{r}}(\boldsymbol{\textit{T}},\boldsymbol{\textit{x}}_{\boldsymbol{\text{0}}})\boldsymbol{\text{<}}\boldsymbol{\textit{w}}_{\boldsymbol{\textit{a}}}(\boldsymbol{\textit{T}})$（其中$\boldsymbol{\textit{r}}(\boldsymbol{\textit{T}},\boldsymbol{\textit{x}}_{\boldsymbol{\text{0}}})$为$\boldsymbol{\textit{T}}$在$\boldsymbol{\textit{x}}_{\boldsymbol{\text{0}}}$处的局部谱半径），则$\boldsymbol{\text{闭包}}\boldsymbol{\text{\textbar{}}}\boldsymbol{\textit{x}}\boldsymbol{\text{∈}}\boldsymbol{\textit{H}}\boldsymbol{\text{:}}\boldsymbol{\textit{r}}(\boldsymbol{\textit{T}},\boldsymbol{\textit{x}})\boldsymbol{\text{≤}}\boldsymbol{\textit{r}}(\boldsymbol{\textit{T}},\boldsymbol{\textit{x}}_{\boldsymbol{\text{0}}})\boldsymbol{\text{\textbar{}}}$是$\boldsymbol{\textit{T}}$的非平凡超不变子空间，$\boldsymbol{\text{闭包}}\boldsymbol{\text{span}}\boldsymbol{\text{\textbar{}}}\boldsymbol{\textit{T}}^{\boldsymbol{\textit{n}}}\boldsymbol{\textit{x}}_{\boldsymbol{\text{0}}}\boldsymbol{\text{:}}\boldsymbol{\textit{n}}\boldsymbol{\text{≥}}\boldsymbol{\text{0}}\boldsymbol{\text{\textbar{}}}$是$\boldsymbol{\textit{T}}$的非平凡不变子空间；还证明一般情况下$\boldsymbol{\textit{w}}_{\boldsymbol{\textit{a}}}(\boldsymbol{\textit{T}})\boldsymbol{\text{≤}}\boldsymbol{\textit{r}}(\boldsymbol{\textit{T}})$（$\boldsymbol{\textit{r}}(\boldsymbol{\textit{T}})$为$\boldsymbol{\textit{T}}$的谱半径），且对所有亚正规算子有$\boldsymbol{\textit{w}}_{\boldsymbol{\textit{a}}}(\boldsymbol{\textit{T}})\boldsymbol{\text{=}}\boldsymbol{\textit{r}}(\boldsymbol{\textit{T}})$。此外，对任意$\boldsymbol{\textit{x}}\boldsymbol{\text{∈}}\boldsymbol{\textit{H}}$，$\boldsymbol{\textit{W}}_{\boldsymbol{\textit{a}}}(\boldsymbol{\textit{T}},\boldsymbol{\textit{x}})$是紧区间，$\boldsymbol{\textit{W}}_{\boldsymbol{\textit{a}}}(\boldsymbol{\textit{T}})$是有界区间（允许退化为单点）。记$\boldsymbol{\text{弱算子拓扑（WOT）}}$下算子序列$\boldsymbol{\text{\textbar{}}}\boldsymbol{\text{|}}\boldsymbol{\textit{T}}^{\boldsymbol{\textit{n}}}\boldsymbol{\text{|}}^{\boldsymbol{\text{1/}}\boldsymbol{\textit{n}}}\boldsymbol{\text{\textbar{}}}$（$\boldsymbol{\textit{n}}\boldsymbol{\text{=}}\boldsymbol{\text{1}}\boldsymbol{\text{,}}\boldsymbol{\text{2}}\boldsymbol{\text{,}}\boldsymbol{\text{…}}$）的聚点集为$\boldsymbol{\text{}}$，则$\boldsymbol{\textit{W}}_{\boldsymbol{\textit{a}}}(\boldsymbol{\textit{T}},\boldsymbol{\textit{x}})$是$\boldsymbol{\text{WOT}}$连续映射$\boldsymbol{\textit{A}}\boldsymbol{\text{↦}}\boldsymbol{\text{⟨}}\boldsymbol{\textit{A}}\boldsymbol{\textit{x}}\boldsymbol{\text{,}}\boldsymbol{\textit{x}}\boldsymbol{\text{⟩}}$作用在$\boldsymbol{\text{}}$上的像；虽所有此类像均为区间，但$\boldsymbol{\text{}}$本身未必连通：当$\boldsymbol{\textit{T}}$下方有界时$\boldsymbol{\text{}}$连通，一般情况下可能不连通。
英文摘要
For a bounded linear operator $T$ on a complex separable Hilbert space $\mathcal{H}$ and a vector $x\in \mathcal{H}$, let $W_a(T,x)$ be the set of cluster points of the sequence $\{\langle|T^n|^{1/n}x,x\rangle\}_{n=1}^{\infty}$. We define the asymptotic numerical range and the asymptotic numerical radius of $T$, respectively, by $W_a(T):=\bigcup_{\|x\|=1}W_a(T,x)$ and $w_a(T):=\sup W_a(T)$. We prove that $W_a(T,x)$ is a compact interval for every $x\in\mathcal{H}$ and that $W_a(T)$ is a bounded interval, where intervals are allowed to be degenerate. Using the asymptotic numerical radius, we show that if there is a nonzero vector $x_0\in \mathcal{H}$ with $r(T,x_0)<w_a(T)$, where $r(T,x_0)$ denotes the local spectral radius of $T$ at $x_0$, then the subspaces $\overline{\{x\in\mathcal{H}:r(T,x)\le r(T,x_0)\}}$ and $\overline{\operatorname{span}\{T^n x_0:n\ge0\}}$, which are well known to be hyperinvariant and invariant for $T$, respectively, are both nontrivial. We also prove that $w_a(T)=r(T)$ for every hyponormal operator $T$, where $r(T)$ denotes the spectral radius of $T$. Finally, we investigate the connectedness of the set of WOT cluster points of the sequence $\{|T^n|^{1/n}\}_{n=1}^\infty$.
