A Note on Weak Hypercyclicity and Linear Fractional Composition Operator

TL;DR

This paper introduces a new criterion for weak hypercyclicity of operators, answers longstanding questions, and shows conditions under which weak hypercyclicity and hypercyclicity coincide for certain composition operators.

Contribution

It establishes a weak hypercyclicity criterion, resolves open questions from 2004, and demonstrates equivalence of weak and strong hypercyclicity for specific composition operators.

Findings

Established a sufficient condition for weak hypercyclicity.

Answered open questions 5.3 and 5.8 from Chan and Sanders (2004).

Proved that for certain composition operators, weak and hypercyclicity are equivalent.

Abstract

This paper discusses the existence of a sufficient condition for an operator to be weakly hypercyclic. We establish a weak hypercyclicity criterion, and thereupon we can answer questions 5.3 and 5.8 posed by Chan and Sanders in 2004. Lastly, we show that for specific type of composition operator, weak hypercyclicity and hypercyclicity are equivalent.