Commutator criteria for strong mixing

We present new criteria, based on commutator methods, for the strong mixing property of discrete flows $\{U^N{\}}_{N\in \mathbb{Z}}$ and continuous flows $\{{\mathrm{e}}^{-itH}{\}}_{t\in \mathbb{R}}$ induced by unitary operators $U$ and self-adjoint operators $H$ in a Hilbert space $\mathcal{H}$. Our approach put into evidence a general definition for the topological degree of the curves $N\mapsto U^N$ and $t\mapsto {\mathrm{e}}^{-itH}$ in the unitary group of $\mathcal H$. Among other examples, our results apply to skew products of compact Lie groups, time changes of horocycle flows and adjacency operators on graphs.