AI 中文总结
研究巴拿赫空间上拟幂零算子的幂集，证明对任意巴拿赫空间上的拟幂零算子\(1\in\Lambda(T)\)，还表明\(\ell^p\)上特定后向单侧加权移位的\(\Lambda(T)=[0,1]\)，回答了相关问题并弱化了前人假设。
AI 中文摘要
对于巴拿赫空间\(X\)上的拟幂零算子\(T\)，R. 道格拉斯和R. 杨为每个非零向量\(x\)关联局部预解式增长指数\(k_x\)，并引入幂集\(\Lambda(T)=\{k_x:x\neq0\}\)。我们证明对于任意巴拿赫空间上的每个拟幂零算子，\(1\in\Lambda(T)\)，这回答了季和张的一个问题。我们进一步表明，对于\(\ell^p\)上权重序列严格递减且对某个\(p'\gt0\)是\(p'\)可和的每个后向单侧加权移位，\(\Lambda(T)=[0,1]\)，从而弱化了胡和季所施加的假设。
英文摘要
For a quasinilpotent operator $T$ on a Banach space $X$, R. Douglas and R. Yang associated with each nonzero vector $x$ the local resolvent-growth exponent $k_x$, and introduced the power set $Λ(T) = \{k_x : x \neq 0\}$. We prove that $1 \in Λ(T)$ for every quasinilpotent operator on an arbitrary Banach space, which answers a question of Ji and Zhang. We further show that $Λ(T) = [0,1]$ for every backward unilateral weighted shift on $\ell^p$ whose weight sequence is strictly decreasing and $p'$-summable for some $p' > 0$, thereby weakening the hypotheses imposed by Hu and Ji.
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