Existence of common and upper frequently hypercyclic subspaces

TL;DR

This paper establishes criteria for the existence of common and upper frequently hypercyclic subspaces, demonstrating their presence in various operators including differentiation and weighted shift operators, and answering a notable open question.

Contribution

It introduces new criteria for hypercyclic subspaces and applies them to specific operators, expanding understanding of hypercyclicity in functional analysis.

Findings

Existence of operators with upper frequently hypercyclic subspaces but no frequently hypercyclic subspace.

Differentiation operators on entire functions support upper frequently hypercyclic subspaces.

A family of scalar multiples of differentiation operators has a common hypercyclic subspace.

Abstract

We provide criteria for the existence of upper frequently hypercyclic subspaces and for common hypercyclic subspaces, which include the following consequences. There exist frequently hypercyclic operators with upper-frequently hypercyclic subspaces and no frequently hypercyclic subspace. On the space of entire functions, each differentiation operator induced by a non-constant polynomial supports an upper frequently hypercyclic subspace, and the family of its non-zero scalar multiples has a common hypercyclic subspace. A question of Costakis and Sambarino on the existence of a common hypercyclic subspace for a certain uncountable family of weighted shift operators is also answered.