A dynamical proof of the van der Corput inequality. Issue 4 (2nd October 2022)

• Record Type: Journal Article
• Authors: Edeko, Nikolai; Kreidler, Henrik; Nagel, Rainer
• Is Part Of: Dynamical systems. Volume 37:Issue 4(2022)
• Page Start: 648
• Page End: 665
• Publication Date: 2022-10-02
• Subjects: Ergodic theorems -- Furstenberg correspondence principle -- C*-algebras -- Hilbert modules; Primary: 47A35 -- Secondary: 46L08 -- 46L55; Differentiable dynamical systems -- Periodicals; 515.35205
• DOI: 10.1080/14689367.2022.2100244
• Languages: English
• ISSNs: 1468-9367

Abstract

Abstract : We provide a dynamical proof of the van der Corput inequality for sequences in Hilbert spaces that is based on the Furstenberg correspondence principle. This is done by reducing the inequality to the mean ergodic theorem for contractions on Hilbert spaces. The key difficulty therein is that the Furstenberg correspondence principle is, a priori, limited to scalar-valued sequences. We, therefore, discuss how interpreting the Furstenberg correspondence principle via the Gelfand–Naimark–Segal construction for C ∗ -algebras allows to study not just scalar but general Hilbert space-valued sequences in terms of unitary operators. This yields a proof of the van der Corput inequality in the spirit of the Furstenberg correspondence principle and the flexibility of this method is discussed via new proofs for different variants of the inequality.