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On frequently supercyclic operators and an F_Γ-hypercyclicity criterior with applications

Thiago R. Alves, Geraldo Botelho, Vinicius V. Fávaro

Abstract

Given a Furstenberg family F and a subset Γ of C, we introduce and explore the notions of F_Γ-hypercyclic operator and F-hypercyclic scalar set. First, the study of F_C-hypercyclic operators yields new interesting information about frequently supercyclic, U-frequently supercyclic, reiteratively supercyclic and supercyclic operators. Then we provide a criterion for identifying F_Γ-hypercyclic operators. As applications of this criterion, we show that any unilateral pseudo-shift operator on c_0(N) or l_p(N) is F_Γ-hypercyclic for every unbounded subset Γ of C. Moreover, under the same condition on Γ, we show that any separable infinite-dimensional Banach space supports an F_Γ-hypercyclic operator. Finally, our study provides sufficient and necessary conditions for a subset Γ of C to be a hypercyclic scalar set. These results give partial answers to a question raised by Charpentier, Ernst, and Menet in 2016.

On frequently supercyclic operators and an F_Γ-hypercyclicity criterior with applications

Abstract

Given a Furstenberg family F and a subset Γ of C, we introduce and explore the notions of F_Γ-hypercyclic operator and F-hypercyclic scalar set. First, the study of F_C-hypercyclic operators yields new interesting information about frequently supercyclic, U-frequently supercyclic, reiteratively supercyclic and supercyclic operators. Then we provide a criterion for identifying F_Γ-hypercyclic operators. As applications of this criterion, we show that any unilateral pseudo-shift operator on c_0(N) or l_p(N) is F_Γ-hypercyclic for every unbounded subset Γ of C. Moreover, under the same condition on Γ, we show that any separable infinite-dimensional Banach space supports an F_Γ-hypercyclic operator. Finally, our study provides sufficient and necessary conditions for a subset Γ of C to be a hypercyclic scalar set. These results give partial answers to a question raised by Charpentier, Ernst, and Menet in 2016.

Paper Structure

This paper contains 7 sections, 20 theorems, 83 equations.

Table of Contents

  1. Introduction and overview of the main results
  2. The inclusions in (\ref{['inclusions']}) are all strict
  3. The $\mathscr{F}_\Gamma$-hypercyclicity criterion
  4. Applications of the $\mathscr{F}_\Gamma$-hypercyclicity criterion
  5. Unilateral pseudo-shift operators
  6. Existence of frequently supercyclic operators
  7. A necessary and a sufficient condition for a scalar set to be $\mathscr{F}$-hypercyclic

Key Result

Theorem 1

CharErnMenet16 A non-empty subset $\Gamma$ of $\mathbb{C}$ is a hypercyclic scalar set if and only if $\Gamma \setminus \{0\}$ is non-empty, bounded, and bounded away from zero.

Theorems & Definitions (43)

  • Theorem
  • Definition 1.1
  • Theorem 2.1
  • proof
  • Corollary 2.2
  • proof
  • Corollary 2.3
  • proof
  • Definition 3.1: BesMenetPerisPuig16
  • Theorem 3.2: $\mathscr{F}_\Gamma$-Hypercyclicity Criterion
  • ...and 33 more