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

Abstract

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 $s\ge 0$. Let $f=\sum _{i=1}^{\infty }{a}_i{v}_i+f^{\perp }$ be the orthogonal decomposition with descending $|a_i|$. We prove that $\underset{\tau \to \infty }{\lim }{\lambda }_{\tau ,i}=|a_i{|}^2$. The continuous one-parameter semigroup $\{{\mathcal{K}}^s:s\ge 0\}$ 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 $X$.

References in corpus (1)

• Eigenvalues of Autocovariance Matrix: A Practical Method to Identify the Koopman Eigenfrequencies