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

Given a stationary continuous-time process f(t), the Hilbert-Schmidt operator Aτ can be defined for every finite τciteVautard1989SingularSA. Let λτ,i be the eigenvalues of Aτ with descending order. In this article, a Hilbert space Hf and the (time-shift) continuous one-parameter semigroup of isometries Ks are defined. Let vi, i∈mathbbN be the eigenvectors of Ks for all s≥ 0. Let f = displaystylesumi=1∞aivi + fperp be the orthogonal decomposition with descending |ai|. We prove that displaystylelimτ→∞λτ,i = |ai|2. The continuous one-parameter semigroup Ks: s≥ 0 is equivalent, almost surely, to the classical Koopman one-parameter semigroup defined on L2(X,nu), if the dynamical system is ergodic and has invariant measure nu on the phase space X.