华东师范大学学报(自然科学版) ›› 2012, Vol. 2012 ›› Issue (1): 106-112, 120.

• 应用数学与基础数学 • 上一篇    下一篇

子空间超循环性与公共的子空间超循环向量

赵显锋, 舒永录, 周云华   

  1. 重庆大学 数学与统计学院, 重庆 401331
  • 收稿日期:2011-04-01 修回日期:2011-07-01 出版日期:2012-01-25 发布日期:2012-01-26

Subspace-supercyclicity and common subspace-supercyclic vectors

ZHAO Xian-feng, SHU Yong-lu, ZHOU Yun-hua   

  1. College of Mathematics and Statistics, Chongqing University, Chongqing 401331,  China
  • Received:2011-04-01 Revised:2011-07-01 Online:2012-01-25 Published:2012-01-26

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摘要： 若无限维可分的~Banach~空间上的线性有界算子~$T$~满足: 对某个非零子空间~$M$, 存在向量~$x$~使~$\mathbb{C}\cdot O(x, T)\bigcap M$~在~$M$~中稠密, 则称~$T$~是子空间超循环算子. 构造例子说明了子空间超循环性并非是无限维现象, 以及子空间超循环算子并不一定是超循环的; 同时, 还给出了一个子空间超循环准则和一族算子的公共的子空间亚超循环(子空间超循环) 向量是稠密~$G_\delta$~集的充要条件.

关键词: 子空间超循环性, 公共的子空间亚超循环向量, 公共的子空间超循环向量

Abstract: A bounded linear operator $T$ on Banach space is subspace-supercyclic for a nonzero subspace $M$ if there exists a vector whose projective orbit intersects the subspace $M$ in a relatively dense set. We constructed examples to show that subspace-supercyclic is not a strictly infinite dimensional
phenomenon, and that some subspace-supercyclic operators are not supercyclic. We provided a subspace-supercyclicity criterion and offered two necessary and sufficient conditions for a path of bounded linear operators to have a dense $G_\delta$ set of common subspace-hypercyclic vectors and common subspace-supercyclic vectors.

Key words: subspace-supercyclicity, common subspace-hypercyclic vectors, common subspace-supercyclic vectors

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