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Frequently recurrent operators

Antonio Bonilla, Karl-G. Grosse-Erdmann, Antoni López-Martínez, Alfred Peris

Abstract

Motivated by a recent investigation of Costakis et al. on the notion of recurrence in linear dynamics, we study various stronger forms of recurrence for linear operators, in particular that of frequent recurrence. We study, among other things, the relationship between a type of recurrence and the corresponding notion of hypercyclicity, the influence of power boundedness, and the interplay between recurrence and spectral properties. We obtain, in particular, Ansari- and Léon-Müller-type theorems for $\mathcal{F}$-recurrence under very weak assumptions on the Furstenberg family $\mathcal{F}$. This allows us, as a by-product, to deduce Ansari- and Léon-Müller-type theorems for $\mathcal{F}$-hypercyclicity.

Frequently recurrent operators

Abstract

Motivated by a recent investigation of Costakis et al. on the notion of recurrence in linear dynamics, we study various stronger forms of recurrence for linear operators, in particular that of frequent recurrence. We study, among other things, the relationship between a type of recurrence and the corresponding notion of hypercyclicity, the influence of power boundedness, and the interplay between recurrence and spectral properties. We obtain, in particular, Ansari- and Léon-Müller-type theorems for -recurrence under very weak assumptions on the Furstenberg family . This allows us, as a by-product, to deduce Ansari- and Léon-Müller-type theorems for -hypercyclicity.

Paper Structure

This paper contains 8 sections, 39 theorems, 78 equations, 1 figure.

Table of Contents

  1. Introduction
  2. Recurrence, hypercyclicity, and the size of the set of recurrent vectors
  3. Recurrence and power boundedness
  4. Recurrence, the unit circle, and the spectrum
  5. Recurrence of weighted backward shift operators
  6. ${\text{IP}}^*$-recurrence
  7. Recurrence and unimodular eigenvectors
  8. $\mathcal{F}$-recurrence

Key Result

Theorem 2.1

Let $T\in L(X)$. Then the following assertions are equivalent: In that case the set of hypercyclic reiteratively recurrent vectors is residual.

Figures (1)

  • Figure 1: Indices for the seminorms $p_n$

Theorems & Definitions (76)

  • Definition 1.1
  • Theorem 2.1
  • proof
  • Corollary 2.2
  • Example 2.3
  • Example 2.4
  • Theorem 2.5
  • proof
  • Example 2.6
  • Theorem 2.7
  • ...and 66 more