A dynamical proof of the van der Corput inequality

TL;DR

This paper offers a novel dynamical proof of the van der Corput inequality for Hilbert space sequences, utilizing the Furstenberg correspondence principle and Gelfand--Naimark--Segal construction to extend to vector-valued sequences.

Contribution

It introduces a dynamical approach to the van der Corput inequality for Hilbert space sequences, broadening the principle's applicability through operator algebra techniques.

Findings

Provides a dynamical proof based on the mean ergodic theorem.

Extends the Furstenberg correspondence principle to vector-valued sequences.

Offers new proofs for various variants of the inequality.

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.