Let $(X, \mathcal X, \mu)$ be a probability space and $T:X\to X$ be an invertible measure preserving transformation. We get a unitary operator $U_T:L^2_\mu\to L^2_\mu$ which takes $f$ to $f\circ T$.
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 $T$.
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 $f\in L^2(\mu)$ of integral zero and a sequence $n_j\uparrow \infty$ such that $$\langle f, T^{n_j}f \rangle\to 1$$ as $j \to \infty$. The function $f$ here is ``essentially'' an eigenvector.