Left Multiplication Mappings on Operator Spaces

Abstract:

      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.