New paper: Uniform convergence in von Neumann’s ergodic theorem in absence of a spectral gap

Jonathan Ben-Artzi and Baptiste Morisse recently submitted a paper entitled Uniform convergence in von Neumann’s ergodic theorem in absence of a spectral gap.

Von Neumann’s ergodic theorem states that “time” averages converge to “spatial” avergaes: given a one-parameter family of unitary maps U_t:\mathcal{H}\to\mathcal{H},\,t\in\mathbb{R}, the average \frac{1}{2T}\int_{-T}^TU_tf\,\mathrm{d}t converges to the projection of f onto the space of functions invariant under U_t as T\to+\infty .

Generally there is no rate. However, if the generator of U_t has a spectral gap, the rate is T^{-1} . In the present paper, it is shown that even in the absence of a spectral gap one can obtain a rate, albeit on a subspace of \mathcal{H} , and with a rate worse than T^{-1} . This is done by obtaining a suitable estimate for the density of the spectrum near zero (low frequencies).

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