to a very simple example of a weakly mixing transformation which is not strongly mixing. This could be of some interest since the classical concrete examples are arguably more complicated (see the one given in [@P section 4.5]).

Regarding the difference between weak and strong mixing, it is also worth pointing out that there exist Hilbert space operators which are weakly mixing in the Gaussian sense but not even strongly mixing in the [topological]{} sense. Indeed, in the beautiful paper [@BadG], C. Badea and S. Grivaux are able to construct a weakly mixing operator (in the Gaussian sense) which is *partially power-bounded*, $\sup_{n\in I} \Vert T^n\Vert<\infty$ for some infinite set $I\subset\NN$. This line of investigations was pursued even much further in [@BadG2] and [@EG].

We have deliberately stressed the formal analogy between weak and strong mixing in the statement of Theorem \[WS\]. In view of this analogy, it should not come as a surprise that Theorem \[WS\] can be deduced from some more general results dealing with abstract notions of “mixing". (In order not to make this introduction exceedingly long, these results will be described in the next section). In particular, (1) and (2) are formal consequences of Theorem \[abstract\] 