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A hypercyclicity criterion for non-metrizable topological vector spaces

Alfred Peris

Abstract

We provide a sufficient condition for an operator $T$ on a non-metrizable and sequentially separable topological vector space $X$ to be sequentially hypercyclic. This condition is applied to some particular examples, namely, a composition operator on the space of real analytic functions on $]0,1[$, which solves two problems of Bonet and Domański \cite{bd12}, and the "snake shift" constructed in \cite{bfpw} on direct sums of sequence spaces. The two examples have in common that they do not admit a densely embedded F-space $Y$ for which the operator restricted to $Y$ is continuous and hypercyclic, i.e., the hypercyclicity of these operators cannot be a consequence of the comparison principle with hypercyclic operators on F-spaces.

A hypercyclicity criterion for non-metrizable topological vector spaces

Abstract

We provide a sufficient condition for an operator on a non-metrizable and sequentially separable topological vector space to be sequentially hypercyclic. This condition is applied to some particular examples, namely, a composition operator on the space of real analytic functions on , which solves two problems of Bonet and Domański \cite{bd12}, and the "snake shift" constructed in \cite{bfpw} on direct sums of sequence spaces. The two examples have in common that they do not admit a densely embedded F-space for which the operator restricted to is continuous and hypercyclic, i.e., the hypercyclicity of these operators cannot be a consequence of the comparison principle with hypercyclic operators on F-spaces.

Paper Structure

This paper contains 2 sections, 2 theorems, 11 equations.

Table of Contents

  1. Criteria for sequential hypercyclicity
  2. Composition operators on the space of real analytic functions and shifts on direct sums

Key Result

Proposition 1

Let $X$ be a sequentially separable topological vector space and $T\in L(X)$ such that there exist a sequentially dense set $X_0:=\{ x_n \ ; \ n\in \mathbb{N}\}\subset X$, a sequence of maps $S_n:X_0\rightarrow X$, $n\in\mathbb{N}$, a subspace $Y\subset X$ with a finer topology $\tau$ such that $(Y, Then $T$ is sequentially hypercyclic.

Theorems & Definitions (6)

  • Proposition 1
  • proof
  • Definition 2
  • Corollary 3
  • Example 4
  • Example 5