Logo
ScienceStack
Table of Contents
Fetching ...
Sign In

Quasi-rigid operators and hyper-recurrence

Manuel Saavedra, Manuel Stadlbauer

Abstract

We study recurrent operators from a new perspective by introducing the notion of hyper-recurrent operators and establish robust connections with quasi-rigid operators. For example, we prove that a recurrent operator on a separable Banach space is quasi-rigid if and only if it is a linear factor of a hyper-recurrent operator, and show that the quasi-rigid operators found in Costakis, Manoussos and Parissis's work, along with many others, are, in fact, hyper-recurrent operators. Furthermore, we provide a negative answer, using a class of operators introduced by Tapia, to the question by Costakis et al. whether $T \oplus T$ is recurrent whenever $T$ is.

Quasi-rigid operators and hyper-recurrence

Abstract

We study recurrent operators from a new perspective by introducing the notion of hyper-recurrent operators and establish robust connections with quasi-rigid operators. For example, we prove that a recurrent operator on a separable Banach space is quasi-rigid if and only if it is a linear factor of a hyper-recurrent operator, and show that the quasi-rigid operators found in Costakis, Manoussos and Parissis's work, along with many others, are, in fact, hyper-recurrent operators. Furthermore, we provide a negative answer, using a class of operators introduced by Tapia, to the question by Costakis et al. whether is recurrent whenever is.
Paper Structure (12 sections, 34 theorems, 84 equations, 1 table)

This paper contains 12 sections, 34 theorems, 84 equations, 1 table.

Table of Contents

  1. Introduction
  2. Preliminaries
  3. Quasi-rigid operatores
  4. Augé-Tapia operators
  5. Hyper-recurrent operators
  6. Stationary sequences
  7. Hyper-recurrent operators
  8. Exploring hyper-recurrence
  9. Power bounded operators
  10. Multiplication Operators
  11. Operators with discrete spectrum
  12. Composition operators on spaces of holomorphic functions

Key Result

Proposition 2.2

Let $T$ be a recurrent operator on a Fréchet space $X$. Then $\mathfrak{L}(\omega)$ and $\overline{\mathfrak{L}(\omega)}$ are $T$-invariant linear subspaces, $\mathfrak{L}(\sigma(\omega))=\mathfrak{L}(\omega)$ and $\mathfrak{L}(\omega) \subset \mathfrak{L}(\mu)$ for any subsequence $\mu$ of $\omega\

Theorems & Definitions (81)

  • Definition 2.1
  • Proposition 2.2
  • proof
  • Proposition 2.3
  • proof
  • Definition 3.2
  • Theorem 3.3
  • Theorem 3.4: Mycielski Theorem, Corollary 1.1 in Tan
  • proof : Proof of Theorem \ref{['equiv']}
  • Remark 3.5
  • ...and 71 more