On hypercyclic spaces and (common) $\mathscr{U}$-frequently hypercyclic spaces

TL;DR

This paper investigates the existence of hypercyclic and $$-frequently hypercyclic subspaces in unilateral weighted backward shift operators on $$-space, providing new results and solving an open problem from 2015.

Contribution

It proves the existence of hypercyclic and $$-frequently hypercyclic subspaces free of certain vectors, and addresses an open question on common $$-frequently hypercyclic subspaces.

Findings

Existence of $$-frequently hypercyclic subspaces without hypercyclic vectors.

Existence of hypercyclic subspaces free of $$-frequently hypercyclic vectors.

Resolution of a 2015 open problem on common $$-frequently hypercyclic subspaces.

Abstract

Let $B$ be an unilateral weighted backward shift on $\ell_p$, $1 \leq p < \infty$, that admits a $\mathscr{U}$-frequently hypercyclic subspace. We prove that $B$ admits such a subspace free of frequently hypercyclic vectors. The proof technique we develop also allows us to prove that $B$ admits a hypercyclic subspace free of $\mathscr{U}$-frequently hypercyclic vectors, and to solve a question posed by B\`es and Menet in 2015 on the existence of common $\mathscr{U}$-frequently hypercyclic subspaces.