Frequent universality criterion and densities

TL;DR

This paper enhances the frequent universality and hypercyclicity criteria by employing sharper weighted densities, and constructs an operator that is logarithmically-frequently hypercyclic but not frequently hypercyclic.

Contribution

It provides the optimal conclusion for the criteria using A-densities and introduces a new operator with unique hypercyclicity properties.

Findings

Improved the frequent universality criterion with sharper densities

Constructed an operator that is logarithmically-frequently hypercyclic but not frequently hypercyclic

Established the optimal conclusion for the criteria using A-densities

Abstract

We improve a recent result by giving the optimal conclusion possible both to the frequent universality criterion and the frequent hypercyclicity criterion using the notion of A-densities, where A refers to some weighted densities sharper than the natural lower density. Moreover we construct an operator which is logarithmically-frequently hypercyclic but not frequently hypercyclic.