A semigroup analogue of the Fonf--Lin--Wojtaszczyk ergodic characterization of reflexive Banach spaces with a basis

TL;DR

The paper develops semigroup ergodic characterizations for Banach spaces with a basis, paralleling the Fonf–Lin–Wojtaszczyk results for operators. It constructs bounded uniformly continuous semigroups from Schauder decompositions to obtain a semigroup analogue of FLW01 Cor. 3, showing mean ergodicity alone does not imply uniform mean ergodicity in infinite dimensions. The main result proves that, for spaces with a basis, reflexivity is equivalent to every bounded strongly continuous semigroup being mean ergodic, and equivalently to every bounded uniformly continuous semigroup being mean ergodic. The work also offers a Nagel-based alternative proof and an explicit l^1 example to illustrate the constructions and highlights the distinct roles of semigroup mean ergodicity vs generator properties.

Abstract

In analogy to a recent result by V. Fonf, M. Lin, and P. Wojtaszczyk, we prove the following characterizations of a Banach space $X$ with a basis. (i) $X$ is finite-dimensional if and only if every bounded, uniformly continuous, mean ergodic semigroup on $X$ is uniformly mean ergodic. (ii) $X$ is reflexive if and only if every bounded strongly continuous semigroup is mean ergodic if and only if every bounded uniformly continuous semigroup on $X$ is mean ergodic.

Theorems & Definitions (17)

• Definition 2.1
• Lemma 2.2
• Remark 2.3
• Lemma 2.4
• Lemma 2.5
• proof
• Theorem 2.6
• proof
• Lemma 3.1
• Remark 3.2