We study several notions of boundedness for operators. It is known that any power bounded operator is absolutely Cesàro bounded and strongly Kreiss bounded (in particular, uniformly Kreiss bounded). The converses do not hold in general. In this note, we give examples of topologically mixing (hence, not power bounded) absolutely Cesàro bounded operators on ℓ p (ℕ), 1 ≤ p < ∞, and provide examples of uniformly Kreiss bounded operators which are not absolutely Cesàro bounded. These results complement a few known examples (see [27] and [2]). We also obtain a characterization of power bounded operators which generalizes a result of Van Casteren [32]. In [2] Aleman and Suciu asked if every uniformly Kreiss bounded operator T on a Banach space satisfies that $${\lim _{n \to \infty }}\left\| {{{{T^n}} \over n}} \right\|\; = \;0.$$ . We solve this question for Hilbert space operators and, moreover, we prove that, if T is absolutely Cesàro bounded on a Banach (Hilbert) space, then ∥ T n ∥ = o ( n ) ( $$(\left\| {{T^n}} \right\| = o({n^{{1 \over 2}}}),$$ ( ∥ T n ∥ = o ( n 1 2 ) , , respectively). As a consequence, every absolutely Cesàro bounded operator on a reflexive Banach space is mean ergodic.

中文翻译：

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Banach空间中的Cesàro有界算子

我们研究了运算符有界的几个概念。众所周知，任何幂有界算子都是绝对 Cesàro 有界和强 Kreiss 有界的（特别是一致 Kreiss 有界）。对话一般不成立。在本笔记中，我们给出了在 ℓ p (ℕ), 1 ≤ p < ∞ 上拓扑混合（因此不是幂有界）绝对 Cesàro 有界算子的示例，并提供了不是绝对 Cesàro 有界的均匀 Kreiss 有界算子的示例。这些结果补充了一些已知的例子（见 [27] 和 [2]）。我们还获得了对范卡斯特伦 [32] 的结果进行概括的幂有界算子的特征。在 [2] 中，Aleman 和 Suciu 询问 Banach 空间上的每个一致 Kreiss 有界算子 T 是否满足 $${\lim _{n \to \infty }}\left\| {{{{T^n}} \over n}} \right\|\; = \;0.$$ 。