A Note on Topological and Strict Transitivity

Let \(X\) be a normed linear space and \(L(X)\) be the algebra of continuous linear operators on \(X\). We give a necessary condition for topological transitivity of subsets of \(L(X)\) which gives a necessary condition for hypercyclicity and supercyclicity of a single operator on \(X\). Also, we prove the strict transitivity of some particular families of operators on locally convex spaces and Hilbert spaces.