$T$ is strongly mixing in the Gaussian sense.

3.  In [(1)]{} and [(2)]{}, the converse implications are true if $X$ is a Banach space with [cotype 2]{}.

Some remarks are in order regarding the scope and the “history" of these results.

1.  When $X$ is a Hilbert space, (1) is stated in [@F] (with some additional assumptions on the operator $T$) and a detailed proof is given in [@BG3] (without these additional assumptions). The definition of “perfectly spanning" used in [@BG3] is formally stronger than the above one, but the two notions are in fact equivalent ([@Sophie]).

2.  It is shown in [@Sophie] that under the assumption of (1), the operator $T$ is frequently hypercyclic. The proof is rather complicated, and it is not clear that it could be modified to get weak mixing in the Gaussian sense. However, some of the ideas of [@Sophie] will be crucial for us. In particular, sub-section \[Sophiesection\] owes a lot to [@Sophie].

3.  In the weak mixing case, (3) is proved in [@BG2 Theorem 4.1].

4.  It seems unnecessary to recall here the definition of cotype (see any book on Banach space theory, [@AK]). Suffices it to say that this is a geometrical 