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 $X$.


Introduction
Let {f (t) ∈ C : t ≥ 0} be a continuous time process. We assume that f has zero temporal mean and the lagged moments exist for all s ≥ 0: Define ρ −s =ρ s . In [3] the self-adjoint operator A τ is defined to act on L 2 ([0, τ ]): for every g ∈ L 2 ([0, τ ]), and for all t ∈ [0, τ ]. When ρ ∈ L 2 loc (R) and ρ(s) = 0 for almost all s ∈ [0, τ ] , A τ is a Hilbert-Schmidt operator. In particular, A τ is compact and always has a purely punctual spectrum. As stated in [3], the singular spectrum analysis (SSA) algorithm is based on the spectral analysis of A τ .
While in practice the SSA method has been applied successfully to a large variety of time series, in a theoretical purpose, yet with practical consequences, one may ask ourselves what is the relation between A τ1 and A τ2 for different τ 1 and τ 2 ? And what is the asymptotic behavior of A τ as τ → ∞? In what way is the spectral property of A τ related to intrinsic properties of the dynamical system? In this article we generalize the idea and tools developed in [4] and apply them to study of A τ . We shall prove that where λ τ,i is the i−th largest eigenvalue of A τ and a i is the i−th largest (in modulus) coefficient of some eigenvector v i (of unit length) of the time-shift operator K s (for all s ≥ 0) in the orthogonal decomposition of f : where f ⊥ denotes the the expression of f in the orthogonal complement of the space spanned by the time-shift operator eigenfunctions. If there are only finitely many i (say only N terms in the summation) in Eq.(4), then we set a i = 0 for i > N . The time-shift operator K s is closely related to the classical Koopman operator, which is defined to act, as a time-shift operator, on some function space whose domain is the whole phase space of the dynamical system. In section 2 we present the main result and a brief introduction of the mathematical tools used by the proof of the main result. All the quantities mentioned above are defined rigorously in section 2. The detailed proof of the main result is presented in section 3.
Notes and Comments. The main result as well as the techniques and ideas used for the proof are close in spirit to those developed in [4]. However, the Hilbert-Schmidt operator A τ is defined for continuous time process and the theory developed in [4] does not cover the continuous-time case. The objective of this paper is to confirm that the asymptotic behavior of the Hilbert-Schmidt operator A τ is well related to Koopman theory.

Preliminaries and the main result
Let {f (t) : t ≥ 0} be a continuous-time process.
and that ρ(s) is well-defined by Eq.(1) for all s ≥ 0.
For any s ≥ 0, we use F s to denote the time series {F s (t) = f (t + s) : t ≥ 0}. For any two time series g = {g(t) : t ≥ 0} and h = {h(t) : t ≥ 0}, we define the new time series where a, b ∈ C. We consider the following linear space: Each element h ∈ H s can be written as for any n ≥ 1, c i ∈ C, s i ≥ 0. The existence of ρ(s) allows us to define the following positive semi-definite Hermitian form: Since the Hermitian form is positive semi-definite, V is a linear subspace ofH f . And the Hermitian form is strictly positive-definite on the quotient space H f /V . Hence it defines an inner product on H f /V . We define where the closure is taken with respect to the inner product defined above. We define the operator K s on H f for any s, s 1 ≥ 0: It is obvious that for any h, g ∈H f and any s ≥ 0. Hence K s is well-defined on H f /V , and can be further extended to the whole H f by continuity. Therefore we obtain a one parameter family of isometric operators K s that acts on the Hilbert space H f . And obviously we have To simplify the notation, we use f to also denote the continuous-time process F 0 . We further assume that In other words, assumption 2 assumes that the curve: for all g ∈ H, x ∈ R, and s ≥ 0.

Theorems 1 and 2 suggest the orthogonal decomposition
and where a i 's are the coefficients of the eigenvectors in the following decomposition: We rearrange the index of v i so that |a 1 | ≥ |a 2 | ≥ · · · ≥ 0. In order to make connection with A τ , we need the following lemmas.
Lemma 1. For any τ > 0 and any g ∈ L 2 ([0, τ ]), the following integral is well-defined and is an element of H f .
The proof of this and the following lemma use standard argument from mathematical analysis and we leave the proof to the interested readers. Let H int f is a linear subspace of H f . We have For simplicity, we use the notation L 2 τ := L 2 ([0, τ ]). Given lemma 1, for any Cauchy-Schwartz inequality implies that where , L 2 τ refers to the inner product in L 2 τ and , H f refers to the inner product in H f . Therefore Riesz representation theorem warrants that there exists a linear bounded operator A τ : which is the same as the definition of A τ in [3]. Assumption 2 implies that ρ ∈ L 2 loc (R). This implies that A τ is a Hilbert-Schmidt operator on L 2 τ . We shall use the following variational description of the eigenvalues.
Proposition 1 (The min-max principle). Let H be a Hilbert space and A a Hermitian operator on H. Let λ 1 ≥ λ 2 ≥ · · · be the eigenvalues of A in descending order. Then Our main result states that, where λ τ,i stands for the eigenvalues of A τ .
The following proposition [4] demonstrates the correspondence between the eigenfrequencies of the continuous-time time-shift operator and the discrete-time time-shift operator. Please refer to [4] for the notations in the proposition. Proposition 2. Let {f (X t ) : t ≥ 0} be a continuous time process for which ρ s exists for all s ≥ 0. Let ∆t > 0 be a time step. Assume that for all s ≥ 0.
Remark 1. It is worth to point out that the one-parameter semigroup of isometries {K s : s ≥ 0} is equivalent to the classical Koopman one-parameter semigroup {K s : s ≥ 0} which acts on L 2 (X, dν) almost surely (with respect to the initial state of the time series), if the dynamical system is ergodic and has finite invariant measure ν on the phase space X. Because if f ∈ L 2 (X, ν), then fK sf ∈ L 1 (X, dν) and Birkhoff ergodic theorem states that ρ(s) = ν(fK sf ) for almost every initial state x 0 ∈ X. In other words, f, . Note that f is interpreted as a given time series on the left of the equality and interpreted as a function on the right of the equality. This shows that under the assumption that the dynamical system is ergodic and (finite) measure-preserving, there is a natural isometric bijection from H f to L 2 (X, dν).
For mathematical interests, we present the main result in an abstract mathematical form. Then A f,τ is a Hilbert-Schmidt operator and hence has purely punctual spectrum. Let λ f,τ,i be the i−th largest eigenvalue of A f,τ . Then we have

Proof of the main result
For any fixed small ǫ ≥ 0, we choose k, so that  Let λ τ d ,k,i , λ τ,d,ǫ,i , λ τ,c,i , and λ τ,N U,i be the i−th largest eigenvalue of A τ,d,k , A τ,d,ǫ , A τ,c , A τ,N U respectively. We will prove the following identities: λ τ,d,ǫ,1 ≤ ǫ for any τ > 0, Before we start to prove Eqs.(37)-(40), it is not hard to see that proposition 1 and proposition 3 directly implies the main result. Indeed, for any fixed n and any ǫ > 0, we can find k so that n ≤ k and ∞ i=k+1 |a i | 2 ≤ ǫ. Then we find τ large enough so that λ τ,c,1 ≤ ǫ and λ τ,N U,1 ≤ ǫ. Note that A τ,d,k , A τ,d,ǫ , A τ,c , and A τ,N U are all positive semi-definite. Applying the min-max principle we have and that Combined with Eq.(37), this implies Theorem 3.