Topologically mixing hypercyclic operators

Let $X$ be a separable Fréchet space. We prove that a linear operator $T:X\to X$ satisfying a special case of the Hypercyclicity Criterion is topologically mixing, i.e. for any given open sets $U,V$ there exists a positive integer $N$ such that $T^n(U)\cap V\neq \emptyset$ for any $n\ge N.$ We also characterize those weighted backward shift operators that are topologically mixing.