of $T$ and the rotational invariance of the Gaussian variables $g_n$ ($\mu\circ T^{-1}\sim\sum_0^\infty g_n T(x_n)=\sum_0^\infty (\lambda_ng_n)\, x_n\sim\sum_0^\infty g_n x_n=\mu$). However, this particular measure $\mu$ cannot be ergodic ([@Sophie]).

Building on Flytzanis’ ideas, the first named author and S. Grivaux came rather close to characterizing the weak and strong mixing properties for Banach space operators in terms of the $\TT$-eigenvectors ([@BG3], [@BG2]). However, this was not quite the end of the story because the sufficient conditions for weak or strong mixing found in [@BG3] and [@BG2] depend on some geometrical property of the underlying Banach space, or on some “regularity" property of the $\TT$-eigenvectors (see the remark just after Corollary \[eigvectfield\]).

In the present paper, our aim is to show that in fact, these assumptions can be completely removed. Thus, we intend to establish “optimal" sufficient conditions for weak and strong mixing in terms of the $\TT$-eigenvectors which are valid on an arbitrary Fréchet space. These conditions turn out to be also necessary when the underlying space $X$ is a Banach space with *cotype 2*, and hence we get complete characterizations of weak and strong mixing in this case. We shall in fact consider some more general notions of “mixing", but 