Spectral Characterization of Strong-Mixing

Question

Let $(X, mathcal X, mu)$ be a probability space and $T:Xto X$ be an invertible measure preserving transformation. We get a unitary operator $U_T:L^2_muto L^2_mu$ which takes $f$ to $fcirc T$.
It is known that

$T$ is ergodic if and only if the only eigenfunctions of $U_T$ correspoding to the eigenvalue $1$ are the constant functions.

$T$ is weak-mixing if and only if the only eigenvalue of $U_T$ is $1$ and the only eigenfunctions are the constant functions.

My question is if there is such a characterization for the strong-mixing property of $T$.

Nishant Chandgotia · Answer

This is perhaps not overtly useful but since you wanted a criterion along the same lines here is something: The transformation $(X, mu, T)$ is not strong mixing if and only if there exists $fin L^2(mu)$ of integral zero and a sequence $n_juparrow infty$ such that
$$langle f, T^{n_j}f rangleto 1$$
as $j to infty$. The function $f$ here is ``essentially'' an eigenvector.

Spectral Characterization of Strong-Mixing

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