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Mean and pointwise ergodicity for composition operators on rearrangement-invariant spaces

Thomas Kalmes, Dalimil Peša

TL;DR

The paper addresses mean and pointwise ergodicity of composition operators $T_\phi$ on rearrangement-invariant spaces, introducing a mild (ACR) condition on the space and its associate and a power-measure-boundedness condition on the symbol to guarantee mean ergodicity. A central innovation is the rearrangement-invariant topology $\xi$, which sits between the norm and weak topologies and yields completeness with dual identification $X'=(X,\xi)^*$ under (ACR). The authors prove mean ergodicity of $T_\phi$ on all r.i. Banach spaces with these properties, derive a pointwise ergodic theorem via a maximal-ergodic inequality, and demonstrate these results for several non-reflexive spaces (e.g., Lorentz and Orlicz classes), providing examples and necessity discussions. Collectively, the work extends ergodic theory for composition operators beyond reflexive settings and offers a robust framework for both norm- and almost-everywhere convergence in a broad class of function spaces.

Abstract

We study ergodicity of composition operators on rearrangement-invariant Banach function spaces. More precisely, we give a natural and easy-to-check condition on the symbol of the operator which entails mean ergodicity on a very large class of rearrangement-invariant Banach function spaces. Further, we present some natural additional assumptions that allow us to obtain pointwise ergodicity. The class of spaces covered by our results contains many non-reflexive spaces, such as the Lorentz spaces $L^{p, 1}$ and $L^{p,\infty}$, $p \in (1, \infty)$, Orlicz spaces $L \log^α L$ and $\exp L^α$, $α> 0$, and the spaces $L^1$ and $L^{\infty}$ over measure spaces of finite measure. The main novelty in our approach is the application of a new locally convex topology which we introduce and which lies strictly between the norm topology and the weak topology induced by the associate space. Throughout, we give several examples which illustrate the applicability of our results as well as highlight the necessity and optimality of our assumptions.

Mean and pointwise ergodicity for composition operators on rearrangement-invariant spaces

TL;DR

The paper addresses mean and pointwise ergodicity of composition operators on rearrangement-invariant spaces, introducing a mild (ACR) condition on the space and its associate and a power-measure-boundedness condition on the symbol to guarantee mean ergodicity. A central innovation is the rearrangement-invariant topology , which sits between the norm and weak topologies and yields completeness with dual identification under (ACR). The authors prove mean ergodicity of on all r.i. Banach spaces with these properties, derive a pointwise ergodic theorem via a maximal-ergodic inequality, and demonstrate these results for several non-reflexive spaces (e.g., Lorentz and Orlicz classes), providing examples and necessity discussions. Collectively, the work extends ergodic theory for composition operators beyond reflexive settings and offers a robust framework for both norm- and almost-everywhere convergence in a broad class of function spaces.

Abstract

We study ergodicity of composition operators on rearrangement-invariant Banach function spaces. More precisely, we give a natural and easy-to-check condition on the symbol of the operator which entails mean ergodicity on a very large class of rearrangement-invariant Banach function spaces. Further, we present some natural additional assumptions that allow us to obtain pointwise ergodicity. The class of spaces covered by our results contains many non-reflexive spaces, such as the Lorentz spaces and , , Orlicz spaces and , , and the spaces and over measure spaces of finite measure. The main novelty in our approach is the application of a new locally convex topology which we introduce and which lies strictly between the norm topology and the weak topology induced by the associate space. Throughout, we give several examples which illustrate the applicability of our results as well as highlight the necessity and optimality of our assumptions.
Paper Structure (22 sections, 76 theorems, 225 equations)

This paper contains 22 sections, 76 theorems, 225 equations.

Table of Contents

  1. Introduction
  2. Preliminaries
  3. Non-increasing rearrangement
  4. Banach function norms and quasinorms
  5. Absolute continuity of the quasinorm
  6. Associate spaces
  7. Classical locally convex topologies on quasi-Banach function spaces
  8. Linear operators on quasi-Banach function spaces
  9. Sublinear operators quasi-Banach function spaces
  10. Fundamental function and Marcinkiewicz endpoint spaces
  11. Composition operators on r.i. quasi-Banach function spaces
  12. Elementary properties of composition operators
  13. Power-boundedness
  14. Absolute continuity of the quasinorm
  15. A rearrangement-invariant topology on r.i. quasi-Banach function spaces
  16. ...and 7 more sections

Key Result

Theorem 1.1

Let $X$ be an r.i. Banach function space over a $\sigma$-finite and resonant measure space $(\mathcal{R},\mu)$, and let $X'$ be the corresponding associate space. Assume that $X$ has absolutely continuous norm and $X'$ has the (ACR) property. Assume that the measurable mapping $\phi \colon \mathcal{ Then the operator $T_{\phi}$ is mean ergodic on $(X, \lVert \cdot \rVert_X)$, i.e. there is some co

Theorems & Definitions (169)

  • Theorem 1.1
  • Theorem 1.2
  • Definition 2.1
  • Definition 2.2
  • Theorem 2.3
  • Definition 2.4
  • Remark 2.5
  • Theorem 2.6
  • Definition 2.7
  • Proposition 2.8
  • ...and 159 more