Suppose $T: X \to X$ has discrete spectrum with $T$ a measure preserving map on a probability space. (Discrete spectrum means its eigenfunctions are orthonormal and span $L^2(X)$). The task is to prove that there is a subsequence $n_k \to \infty$ such that $||T^{n_k}f-f|| \to 0$ for all $f \in L^2(X)$. This is an exercise in Karl Petersen's ergodic theory textbook, and in it he assumes that every measure preserving transformation is invertible.
As it stands I don't think the problem is true, but if we add the assumption that $T$ is ergodic and an invertible measure preserving transformation then I think we have a shot. If we assume these extra conditions then we can say that $T$ is conjugate to an ergodic abelian rotation, call it $S(y)=gy$. Now it becomes a little more clear. Since $S$ is ergodic the orbit $\{a^n \}$ is dense and hence there is a subsequence such that $a^{n_k} \to e$. From this we can say that in the group $S$ converges to the identity in the strong topology then pull back the transformation using the spectral isomorphism.
Is my proof correct, and can we say the same thing if we just assume $T$ has discrete spectrum?
I do not think you need your system to be ergodic.
Let take $f \in L^2(X)$, and pick any $\epsilon>0$. There are a $N$ and functions $g_k$ such that $T g_k = e^{i \theta_k} g_k$ and $$ \| f - \sum_{k=0}^N g_k \| \leq \epsilon /3 $$ Now for every $n$ \begin{align} \| f - T^n f\| &\leq \|f - \sum_{k=0}^N g_k \| + \| \sum_{k=0}^N g_k - \sum_{k=0}^N T^n g_k \| + \| \sum_{k=0}^N T^n g_k - T^n f\| \\ &\leq 2 \epsilon /3 + \sum_{k=0}^N \| g_k - T^n g_k \| \\ &\leq 2 \epsilon /3 + \sum_{k=0}^N \|g_k\| | 1- e^{i n \theta_k}|\leq 2 \epsilon /3 + \sum_{k=0}^N \|g_k\| | \sin(\frac{n \theta_k}{2})| \end{align} Now let's say that the firsts $N_0 \leq N$ $\theta_k$ belong to $\pi \mathbb{Q}$, we can find $q$ such that for every $k\leq N_0$ and every $n$ $$ | \sin(\frac{n q \theta_k}{2})| = 0 $$ The other $\theta_k$ belong to $\pi( \mathbb{R}-\mathbb{Q})$ so you can find a $n_{\epsilon}$ such that for every $N_0 <k \leq N$ $$ | \sin(\frac{n_{\epsilon} q \theta_k}{2})| \leq \frac{\epsilon}{3 N \| g_k\|} $$ For such a $n_\epsilon$ we have $$ \| f - T^{q n_\epsilon} f\| \leq \epsilon $$ decreasing $\epsilon$ lead you to a increasing sequence of integer solving your problem.