Let $(X,\mu)$ be a standard probability space and let $T:X \to X$ be a measure-class preserving transformation such that there is no $T$-invariant measure absolutely continuous with respect to $\mu$. (Such transformations are called type $\mathrm{III}$.) For $1 \leq p < \infty$ let $U_{T,p}$ be the isometry of $L^p(X,\mu)$ given by \begin{equation} (U_{T,p} \cdot f)(x) = \left(\frac{\mathrm{d}T^{-1}\mu}{\mathrm{d}\mu}(x) \right)^{1/p} f(T^{-1}x). \end{equation} A nonzero function $f \in L^1(X,\mu)$ is invariant under $U_{T,1}$ if and only if $\frac{|f|}{||f||_1}$ is the density of a $T$-invariant measure, so by assumption such functions do not exist. My question is whether it is possible that there is a nonzero function $f \in L^2(X,\mu)$ which is invariant under the Koopman operator $U_{T,2}$.

I think this is interesting because if such functions do not exist if and only if the averages \begin{equation} \frac{1}{n} \sum_{k=0}^{n-1} U_{T,2}^k \end{equation} always converge to zero in the strong operator topology.