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Frequently hypercyclic sequences of differential operators on the space of entire functions

L. Bernal-González, M. C. Calderón-Moreno, J. A. Prado-Bassas

Abstract

A criterion to obtain frequent hypercyclicity for a sequence of convolution operators on the space of entire functions on the complex plane is provided. The criterion involves that the generating functions of the operators do not vanish on an appropriate annulus, in the boundary of which the modulus of each term of the sequence is in some sense controlled by the preceding ones or the following ones.

Frequently hypercyclic sequences of differential operators on the space of entire functions

Abstract

A criterion to obtain frequent hypercyclicity for a sequence of convolution operators on the space of entire functions on the complex plane is provided. The criterion involves that the generating functions of the operators do not vanish on an appropriate annulus, in the boundary of which the modulus of each term of the sequence is in some sense controlled by the preceding ones or the following ones.
Paper Structure (4 sections, 4 theorems, 17 equations)

This paper contains 4 sections, 4 theorems, 17 equations.

Table of Contents

  1. Introduction.
  2. Preliminary results.
  3. Frequent hypercyclicity of sequences of differential operators.
  4. Final remarks and questions.

Key Result

Theorem 2.1

Let $X$ be an F-space, $Y$ a separable F-space and $T_n : X \to Y$ a sequence of continuous linear mappings. Suppose that there are a dense subset $Y_0$ of $Y$ and mappings $S_n : Y_0 \to X$ such that, for all $y \in Y_0$, the following hold: Then the sequence $(T_n)$ is frequently hypercyclic.

Theorems & Definitions (6)

  • Theorem 2.1
  • Lemma 2.2
  • Theorem 3.1
  • proof
  • Theorem 3.2
  • proof