Bridging Koopman Operator and Time-Series Auto-Correlation Based Hilbert–Schmidt Operator

Abstract

AbstractGiven a stationary continuous-time process f(t), the Hilbert–Schmidt operator Aτ can be defined for every finite τ. Let λτ,i be the eigenvalues of Aτ with descending order. In this article, a Hilbert space $$\mathcal {H}_f$$ ℋ f and the (time-shift) continuous one-parameter semigroup of isometries $$\mathcal {K}^s$$ K s are defined. Let $$\{v_i, i\in \mathbb {N}\}$$ { v i , i ∈ ℕ } be the eigenvectors of $$\mathcal {K}^s$$ K s for all s ≥ 0. Let $$f = \displaystyle \sum _{i=1}^{\infty }a_iv_i + f^{\perp }$$ f = ∑ i = 1 ∞ a i v i + f ⊥ be the orthogonal decomposition with descending |ai|. We prove that limτ→∞λτ,i = |ai|2. The continuous one-parameter semigroup $$\{\mathcal {K}^s: s\geq 0\}$$ { K s : s ≥ 0 } is equivalent, almost surely, to the classical Koopman one-parameter semigroup defined on L2(X, ν), if the dynamical system is ergodic and has invariant measure ν on the phase space X.