Some questions about subspace-hypercyclic operators

Abstract

A bounded linear operator T on a Banach space X is called subspace-hypercyclic for a subspace M if Orb(T, x) ∩ M is dense in M for a vector x ∈ M . We show examples that answer some questions posed by H. Rezaei [7]. In particular, we provide an example of an operator T such that Orb(T, x) ∩ M is somewhere dense in M , but it is not everywhere dense in M .

Sufficient Conditions for Subspace-Hypercyclicity

“…Jimenez-Munguia, Martinez-Avendano and Peris in [4], construct an Mhypercyclic operator such that for any n ∈ N, neither…”

“…For more information about subspace-hypercyclic operators, one can also see [4] and [8][9] and the references there in. In this paper we state some new sufficient conditions for an operator to be subspace-hypercyclic.…”

线性算子动力系统的研究进展

“…Rezai [92] 证明了对于子空间超循环算子 T 和任意实 (复) 多项式 P , 算子 P (T ) 也有相对稠密的值域, 也提出了许多轨道空间稠密性的问题. 之 后, Jiménez-Munguía 等学者在文献 [93] 中举例说明了存在子空间超循环算子 T , 其轨道是某处稠密 但不是处处稠密的. 关于子空间超循环和子空间亚超循环的其他例子和性质, 可参见文献 [94,95].…”

$J$-Class Semigroup Operators

“…Let T ∈ B(X) and M be a closed subspace of X, we say that T is M -transitive, if for any non-empty open sets U, V in M , there exists n ≥ 0 such that T −n (U ) V contain a non-empty open subset of M .The authors showed that M -transitivity implies M -hypercyclicity. Note that the converse is not true, this is proven recently by C. M. Le in [11]; for more information see [9], [15]. In 2013 S.Talebi, M.Asadipour localized the notion of subspace-transitivity and gave the answer of the question asked by B.F. Madore and R.A. Martnez-Avendano, see [19].…”