Left Multiplication Mappings on Operator Spaces

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 → X.Define a left multiplication mapping LT : B(X) → B(X) by LT (V ) = T V,V ∈ B(X).We investigate the connections between hypercyclic and chaotic behaviors of the left multiplication mapping LT on B(X) and that of operator T on X.We obtain that LT 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 LT is chaotic if and only if T is chaotic in the sense of Devaney.