Bridging Koopman Operator and time-series auto-correlation based Hilbert-Schmidt operator

Given a stationary continuous-time process $f(t)$, the Hilbert-Schmidt operator $A_{\tau}$ can be defined for every finite $\tau$\cite{Vautard1989SingularSA}. Let $\lambda_{\tau,i}$ be the eigenvalues of $A_{\tau}$ with descending order. In this article, a Hilbert space $\mathcal{H}_f$ and the (time-shift) continuous one-parameter semigroup of isometries $\mathcal{K}^s$ are defined. Let $\{v_i, i\in\mathbb{N}\}$ be the eigenvectors of $\mathcal{K}^s$ for all $s\geq 0$. Let $f = \displaystyle\sum_{i=1}^{\infty}a_iv_i + f^{\perp}$ be the orthogonal decomposition with descending $|a_i|$. We prove that $\displaystyle\lim_{\tau\to\infty}\lambda_{\tau,i} = |a_i|^2$. The continuous one-parameter semigroup $\{\mathcal{K}^s: s\geq 0\}$ is equivalent, almost surely, to the classical Koopman one-parameter semigroup defined on $L^2(X,\nu)$, if the dynamical system is ergodic and has invariant measure $\nu$ on the phase space $X$.