  To infer eigenvalues of the infinite-dimensional Koopman operator, we study
the leading eigenvalues of the autocovariance matrix associated with a given
observable of a dynamical system. For any observable $f$ for which all the
time-delayed autocovariance exist, we construct a Hilbert space $\mathcal{H}_f$
and a Koopman-like operator $\mathcal{K}$ that acts on $\mathcal{H}_f$. We
prove that the leading eigenvalues of the autocovariance matrix has one-to-one
correspondence with the energy of $f$ that is represented by the eigenvectors
of $\mathcal{K}$. The proof is associated to several representation theorems of
isometric operators on a Hilbert space, and the weak-mixing property of the
observables represented by the continuous spectrum. We also provide an
alternative proof of the weakly mixing property. When $f$ is an observable of
an ergodic dynamical system which has a finite invariant measure $\mu$,
$\mathcal{H}_f$ coincides with closure in $L^2(X,d\mu)$ of Krylov subspace
generated by $f$, and $\mathcal{K}$ coincides with the classical Koopman
operator. The main theorem sheds light to the theoretical foundation of several
semi-empirical methods, including singular spectrum analysis (SSA),
data-adaptive harmonic analysis (DAHD), Hankel DMD and Hankel alternative view
of Koopman analysis (HAVOK). It shows that, when the system is ergodic and has
finite invariant measure, the leading temporal empirical orthogonal functions
indeed correspond to the Koopman eigenfrequencies. A theorem-based practical
methodology is then proposed to identify the eigenfrequencies of $\mathcal{K}$
from a given time series. It builds on the fact that the convergence of the
renormalized eigenvalues of the Gram matrix is a necessary and sufficient
condition for the existence of $\mathcal{K}-$eigenfrequencies.
