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We provide a dynamical proof of the van der Corput inequality for sequencesin Hilbert spaces that is based on the Furstenberg correspondence principle.This is done by reducing the inequality to the mean ergodic theorem forcontractions on Hilbert spaces. The key difficulty therein is that theFurstenberg correspondence principle is, a priori, limited to scalar-valuedsequences. We therefore discuss how interpreting the Furstenberg correspondenceprinciple via the Gelfand--Naimark--Segal construction for C*-algebras allowsto study not just scalar but general Hilbert space-valued sequences in terms ofunitary operators. This yields a proof of the van der Corput inequality in thespirit of the Furstenberg correspondence principle and the flexibility of thismethod is discussed via new proofs for different variants of the inequality.

A dynamical proof of the van der Corput inequality