Koopman representations for positive definite functions

2023/10/30 by Sohail Farhangi, Farhangi, Sohail
Mathematics · Medicine · #43A35 #Advanced Operator Algebra Research #Dynamical Systems (math.DS) #FOS: Mathematics #Group Theory (math.GR) #Neurological and metabolic disorders #Neurological disorders and treatments

paper · pdf · doi:10.48550/arxiv.2310.19386

openalex publication_date 2023/10/30 · openalex created_date 2025/10/10 · openalex updated_date 2026/07/28

Abstract

We show that for any locally compact second countable group G and any continuous positive definite function ϕ:G→ℂ, there exists an ergodic measure preserving system (X,\mathscrB,μ,\T_g\_{g ∈ G}) and a function f ∈ L²(X,μ) for which ϕ(g) = ⟨ T_gf,f⟩. We also show that if G is a countably infinite abelian group, then there exists a (not necessarily ergodic) measure preserving system (X,\mathscrB,μ,\T_g\_{g ∈ G}) and a function f ∈ L²(X,μ) with |f| = ϕ(0) and ϕ(g) = ⟨ T_gf,f⟩.

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