A Dynamical Proof of the Van der Corput Inequality

The paper’s Theorem 2.2 states exactly the inequality under audit and proves it via an invariant-state/GNS construction and the mean ergodic theorem, culminating in the desired bound (see Theorem 2.2 and its proof sketch employing GNS and ergodic projections, as well as the reduction from the scalar case in Section 1) . The candidate solution establishes the same inequality by a different route: a Hilbert-space van der Corput estimate in each GNS space plus a polarized Schwarz inequality to move T^n inside the state. This argument is substantively correct, with a minor technical omission of an O(∥G∥^2/H) term in the Hilbert-space van der Corput bound which vanishes in the final lim inf over H, so the final inequality remains valid. Hence both are correct, using different proofs.