Left Multiplication Mappings on Operator Spaces

DOI：10.3770/j.issn:1000-341X.2010.03.013

 作者 单位 姚玉武 合肥学院数学与物理系, 安徽 合肥 230601

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$T: X\rightarrow X$所组成的代数.给定$T\in B(X)$,定义一个左乘映射$L_T: B(X)\rightarrow B(X),~L_T(V)=TV,~V \in B(X)$.我们讨论了算子空间$B(X)$上左乘映射$L_T$的超循环和混沌行为与空间$X$上算子$T$的超循环和混沌行为之间的关系,得到$L_T$是强算子拓扑超循环的必要只且要$T$满足超循环标准;如果我们把空间$B(X)$上具有强算子拓扑超循环性和稠密(强算子拓扑意义下)周期点集的算子定义为混沌算子,我们还得到$L_T$是混沌的必要且只要$T$是Devaney意义下混沌的.

Let $X$ be a separable infinite dimensional Banach space and $B(X)$ denote its operator algebra, the algebra of all bounded linear operators $T: X\rightarrow X$. Define a left multiplication mapping $L_T: B(X)\rightarrow B(X)$ by $L_T(V)=TV,~ V \in B(X)$. We investigate the connections between hypercyclic and chaotic behaviors of the left multiplication mapping $L_T$ on $B(X)$ and that of operator $T$ on $X$. We obtain that $L_T$ is SOT-hypercyclic if and only if $T$ satisfies the Hypercyclicity Criterion. If we define chaos on $B(X)$ as SOT-hypercyclicity plus SOT-dense subset of periodic points, we also get that $L_T$ is chaotic if and only if $T$ is chaotic in the sense of Devaney.