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Weak limit semigroup in operator theory and ergodic theory

Tanja Eisner, Valentin Gillet

Abstract

We study the weak limit semigroup of an operator $T$, i.e., the set of all operators being weak limit points of the powers of $T$, in three different but related contexts: Koopman operators of measure-preserving transformations, contractions/isometries/unitaries on separable Hilbert spaces and positive operators on $L^p$-spaces. Hereby we focus on finding large subsets of the weak limit semigroup, in particular in the generic case.

Weak limit semigroup in operator theory and ergodic theory

Abstract

We study the weak limit semigroup of an operator , i.e., the set of all operators being weak limit points of the powers of , in three different but related contexts: Koopman operators of measure-preserving transformations, contractions/isometries/unitaries on separable Hilbert spaces and positive operators on -spaces. Hereby we focus on finding large subsets of the weak limit semigroup, in particular in the generic case.

Paper Structure

This paper contains 23 sections, 47 theorems, 65 equations.

Table of Contents

  1. Introduction
  2. Notation and Preliminaries
  3. Measure-preserving transformations
  4. General Hilbert space operators
  5. Positive operators
  6. First properties of the weak limit semigroup
  7. $\lambda$-Rigidity
  8. Fixed asymptotics and $\lambda$-rigidity: transformations
  9. Fixed asymptotics and $\lambda$-rigidity: operators
  10. Fixed asymptotics and $\lambda$-rigidity: positive operators
  11. A class of examples: operators
  12. A class of examples: transformations
  13. Generic asymptotics for transformations
  14. Typical asymptotics for operators
  15. Typical asymptotics for positive operators
  16. ...and 8 more sections

Key Result

Proposition 2.1

For a contraction $T$ on a Banach space with separable dual, the following assertions hold.

Theorems & Definitions (92)

  • Proposition 2.1: Properties of weak limit semigroup
  • proof
  • Remark 2.2
  • Remark 2.3
  • Lemma 3.1
  • proof
  • Proposition 3.2
  • proof
  • Definition 3.3
  • Theorem 3.4: Stepin Stepin86
  • ...and 82 more