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The frequent hypercyclicity of unbounded operator

Xiongxun Huang, Yonglu Shu

TL;DR

The paper addresses the gap in the theory of frequent hypercyclicity for unbounded operators by developing two unbounded Frequent Hypercyclicity Criteria inspired by Bayart–Grivaux and Bonilla–Grosse-Erdmann. It first extends the criterion to closed densely defined operators using a dense subset $X_0$ and a right-inverse $B$ with unconditional convergence of the series $ extstyle\sum A^n x$ and $ extstyle\sum B^n x$, then generalizes to sequences of operators and to $C$-regularized semigroups, establishing conditions under which the semigroup $ extstyle\{e^{tA} ight ightarrow X}$ is frequently hypercyclic. The work also furnishes concrete examples, including differentiation and weighted backward shifts, demonstrating the practical reach of the criteria in unbounded operator dynamics. Overall, the results broaden the applicability of frequent hypercyclicity to unbounded operators and their semigroups, with implications for differential equations and functional-analytic dynamics.

Abstract

We establish two Frequent Hypercyclicity Criteria for unbounded operators, inspired by the frameworks of Bayart Grivaux and deLaubenfels Emamirad Grosse Erdmann. These criteria simplify the verification and construction of frequently hypercyclic operators.

The frequent hypercyclicity of unbounded operator

TL;DR

The paper addresses the gap in the theory of frequent hypercyclicity for unbounded operators by developing two unbounded Frequent Hypercyclicity Criteria inspired by Bayart–Grivaux and Bonilla–Grosse-Erdmann. It first extends the criterion to closed densely defined operators using a dense subset and a right-inverse with unconditional convergence of the series and , then generalizes to sequences of operators and to -regularized semigroups, establishing conditions under which the semigroup is frequently hypercyclic. The work also furnishes concrete examples, including differentiation and weighted backward shifts, demonstrating the practical reach of the criteria in unbounded operator dynamics. Overall, the results broaden the applicability of frequent hypercyclicity to unbounded operators and their semigroups, with implications for differential equations and functional-analytic dynamics.

Abstract

We establish two Frequent Hypercyclicity Criteria for unbounded operators, inspired by the frameworks of Bayart Grivaux and deLaubenfels Emamirad Grosse Erdmann. These criteria simplify the verification and construction of frequently hypercyclic operators.

Paper Structure

This paper contains 3 sections, 10 theorems, 86 equations.

Table of Contents

  1. Introduction
  2. Closed densely defined operator
  3. Unbounded operator and semi-group

Key Result

Theorem 2.2

Let $A$ be a closed operator on a separable $F$-space $X$ for which $A^r$ is a closed operator for all positive integer $r$. Suppose that there is a dense subset $X_0$ of the domain of $T$ and a mapping $B: X_0 \rightarrow X_0$ such that: Then $A$ is frequently hypercyclic.

Theorems & Definitions (29)

  • Definition 1.1: 1
  • Definition 2.1
  • Theorem 2.2: Frequent Hypercyclicity Criterion
  • Definition 2.3
  • Definition 2.4: bonilla2007frequently
  • Lemma 2.5: bonilla2007frequently
  • Theorem 2.6: Frequent Hypercyclicity Criterion
  • proof
  • Corollary 2.7
  • proof
  • ...and 19 more