Banach空间上拟幂零算子的幂集

吉林大学学报(理学版) ›› 2025, Vol. 63 ›› Issue (1): 15-0023.

Banach空间上拟幂零算子的幂集

胡朝龙, 梁定浩, 纪友清

吉林大学数学学院, 长春 130012

收稿日期:2024-11-26 出版日期:2025-01-26 发布日期:2025-01-26
通讯作者: 纪友清 E-mail:jiyq@jlu.edu.cn

Power Set of Quasinilpotent Operator on Banach Space

HU Chaolong, LIANG Dinghao, JI Youqing

College of Mathematics, Jilin University, Changchun 130012, China

Received:2024-11-26 Online:2025-01-26 Published:2025-01-26

摘要/Abstract

摘要： 令T是无穷维复Banach空间X上的拟幂零算子, 且x∈X＼{0}. 令Λ(T)={k_x: x≠0}, 称为T的幂集. 证明Λ(T)是右闭的, 即对Λ(T)的每个非空有界子集σ有sup σ∈Λ(T). 特别地, 证明对任意无穷维复Banach空间X, 存在X上的拟幂零算子T, 使得Λ(T)=［0,1］.

关键词: 拟幂零算子, 幂集, 右闭性, Schauder基序列

Abstract: Let T be a quasinilpotent operator on an infinite dimensional complex Banach space X and x∈X＼{0}. Let Λ(T)={k_x: x≠0}, and call it the power set of T. We prove that Λ(T) is right closed, that is, sup σ∈Λ(T) for each nonempty bounded subset σ of Λ(T). In particular, we prove that for any infinite dimensional complex Banach space X, there exists a quasinilpotent operator T on X such that Λ(T)=［0,1］.

Key words: quasinilpotent operator, power set, right closed, Schauder basis sequence

中图分类号:

O177.7

胡朝龙, 梁定浩, 纪友清. Banach空间上拟幂零算子的幂集[J]. 吉林大学学报(理学版), 2025, 63(1): 15-0023.

HU Chaolong, LIANG Dinghao, JI Youqing. Power Set of Quasinilpotent Operator on Banach Space[J]. Journal of Jilin University Science Edition, 2025, 63(1): 15-0023.