Abstract Given a stationary continuous-time process f ( t ), the Hilbert–Schmidt operator A τ can be defined for every finite τ . Let λ τ , i be the eigenvalues of A τ 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 | a i |. We prove that lim τ → ∞ λ τ , i = | a i | 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 L 2 ( X , ν ), if the dynamical system is ergodic and has invariant measure ν on the phase space X .