算子空间上的超循环性和混沌的左乘映射(英文)

Hypercyclicity and Chaotic Left Multiplication Mappings on Operator Spaces

设X是一个可分的无限维Banach空间,B(X)表示X的算子代数,即所有有界线性算子T:X→X所组成的代数.给定T∈B(X),定义一个左乘映射L_T:B(X)→B(X),L_T(V)=TV,V∈B(X).我们在算子空间B(X)上给出了一个超循环性标准,并且如果X是一个具有对称基的Banach空间,在它的对偶空间X′上也给出了一个类似的标准.此外,还讨论了算子空间B(X)上左乘映射L_T的超循环性和混沌行为与空间X上的算子T的超循环性和混沌行为之间的关系,得到T是Devaney意义下混沌的必要且只要L_T是混沌的.

Let X be a separable infinite dimensional Banach space and B（X） be the algebra of all bounded linear operators on X. For any T ∈ B（X）, define a left multiplication mapping LT ： B（X） → B（X） by LT（V） = TV, V ∈ B（X）. We give a hypercyclicity criterion on B（X） and, in addition, if X is a Banach space with a symmetric basis we give a similar criterion on its dual space X＇. We also investigate the connections between hypercyclic and chaotic behavior of LT on B（X） and that of T on X, and prove that T is chaotic in the sense of Devaney if and only if the left multiplication operator LT is chaotic.