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Two remarks on the set of recurrent vectors

Antoni López-Martínez, Quentin Menet

TL;DR

The authors construct counterexamples to two open questions on recurrence in linear dynamics: (1) there exist recurrent operators whose set of recurrent vectors $ extup{Rec}(T)$ is not dense lineable, and (2) there exist reiteratively recurrent and cyclic operators whose set of reiteratively recurrent vectors $ extup{RRec}(T)$ is meager. The first result uses a modified biorthogonal-sequence construction with unbounded functionals to ensure a dense yet nonlinearly structured recurrent set, while the second employs upper-triangular two-by-two block operators $T_{oldsymbol{\lambda},oldsymbol{\omega}}$ with carefully chosen diagonal and shift weights to obtain a dense, cyclic operator whose reiterative-recurrence set is meager. Together, these results answer two long-standing questions in the negative and highlight distinct behaviors of recurrence and reiterative recurrence in separable infinite-dimensional Banach spaces.

Abstract

We solve in the negative two open problems, related to the linear and topological structure of the set of recurrent vectors, asked by Sophie Grivaux, Alfred Peris and the first author of this paper. Firstly, we show that there exist recurrent operators whose set of recurrent vectors is not dense lineable; and secondly, we construct operators which are reiteratively recurrent and cyclic, but whose set of reiteratively recurrent vectors is meager.

Two remarks on the set of recurrent vectors

TL;DR

The authors construct counterexamples to two open questions on recurrence in linear dynamics: (1) there exist recurrent operators whose set of recurrent vectors is not dense lineable, and (2) there exist reiteratively recurrent and cyclic operators whose set of reiteratively recurrent vectors is meager. The first result uses a modified biorthogonal-sequence construction with unbounded functionals to ensure a dense yet nonlinearly structured recurrent set, while the second employs upper-triangular two-by-two block operators with carefully chosen diagonal and shift weights to obtain a dense, cyclic operator whose reiterative-recurrence set is meager. Together, these results answer two long-standing questions in the negative and highlight distinct behaviors of recurrence and reiterative recurrence in separable infinite-dimensional Banach spaces.

Abstract

We solve in the negative two open problems, related to the linear and topological structure of the set of recurrent vectors, asked by Sophie Grivaux, Alfred Peris and the first author of this paper. Firstly, we show that there exist recurrent operators whose set of recurrent vectors is not dense lineable; and secondly, we construct operators which are reiteratively recurrent and cyclic, but whose set of reiteratively recurrent vectors is meager.
Paper Structure (9 sections, 7 theorems, 66 equations)

This paper contains 9 sections, 7 theorems, 66 equations.

Table of Contents

  1. Introduction
  2. Notation for a general separable infinite-dimensional Banach space
  3. Dense but not dense lineable sets of recurrent vectors
  4. Constructing the operator $T$
  5. Recurrence properties of $T$
  6. Dense but not co-meager sets of reiteratively recurrent vectors
  7. The family of operators $T_{\boldsymbol{\lambda},\boldsymbol{\omega}}$
  8. Choosing $\boldsymbol{\lambda}=(\lambda_k)_{k\in\mathbb{N}}$ and $\boldsymbol{\omega}=(\omega_k)_{k\in\mathbb{N}}$
  9. Final comments and open problems

Key Result

Theorem 2.1

Let $X$ be any separable infinite-dimensional Banach space. There exists a recurrent operator $T:X\longrightarrow X$ whose set of recurrent vectors $\textup{Rec}(T)$ is not dense lineable.

Theorems & Definitions (17)

  • Theorem 2.1
  • Proposition 2.2
  • proof
  • Proposition 2.3
  • proof
  • Remark 2.4
  • Theorem 3.1
  • Remark 3.2
  • Definition 3.3
  • Lemma 3.4
  • ...and 7 more