LAUSR

Let B(X) be the algebra of bounded linear operators on a Banach space X. A subset E of B(X) is said to be n-SOT dense in B(X) if for every continuous linear operator $$\Lambda $$Λ from B(X) onto $$X^{(n)}$$X(n), the direct sum of n copies of X, $$\Lambda (E)$$Λ(E) is dense in $$X^{(n)}$$X(n). We consider the n-SOT hypercyclic continuous linear maps on B(X), namely, those that have orbits that are n-SOT dense in B(X). Some nontrivial examples of such operators are provided and many of their basic properties are investigated. In particular, we show that the left multiplication operator $$L_T$$LT is 1-SOT hypercyclic if and only if T is hypercyclic on X.