Abstract

We consider the power-law uniform (in the operator norm) convergence on vector subspaces with their own norms in the von Neumann ergodic theorem with discrete time. All possible exponents of the considered power-law convergence are found; for each of these exponents, spectral criteria for such convergence are given and the complete description of all such subspaces is obtained. Uniform convergence on the whole space takes place only in the trivial cases, which explains the interest in uniform convergence precisely on subspaces. In addition, by the way, old estimates of the rates of convergence in the von Neumann ergodic theorem for measure-preserving mappings are generalized and refined.

中文翻译：

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离散时间冯诺依曼遍历定理子空间的一致收敛性

摘要

我们考虑离散时间冯诺依曼遍历定理中具有自己范数的向量子空间上的幂律一致（在算子范数中）收敛。找到所考虑的幂律收敛性的所有可能的指数；对于每个指数，给出了这种收敛的谱标准，并获得了所有此类子空间的完整描述。整个空间上的一致收敛仅发生在微不足道的情况下，这恰恰解释了子空间上一致收敛的兴趣。此外，顺便说一句，对保测映射的冯·诺依曼遍历定理中的收敛率的旧估计进行了概括和改进。