Topological mixing and Hypercyclicity Criterion for sequences of operators

For a sequence $\{T_n\}$ of continuous linear operators on a separable Fréchet space $X$, we discuss necessary conditions and sufficient conditions for $\{T_n\}$ to be topologically mixing, and the relations between topological mixing and the Hypercyclicity Criterion. Among them are: 1) topological mixing is equivalent to being hereditarily densely hypercyclic; 2) the Hypercyclicity Criterion with respect to the full sequence $\mathbb{N}$ implies topological mixing; 3) topological mixing implies the Hypercyclicity Criterion with respect to some sequence $\{n_k\}\subset \mathbb{N}$ that cannot be syndetic in general, and also implies condition (b) of the Hypercyclicity Criterion with respect to the full sequence. Applications to two examples of operators on the Fréchet space $H(\mathbb{C})$ of entire functions are also discussed.