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

Let $B$ be an unilateral weighted backward shift on ${\mathrm{\ell }}_p$, $1\le p<\mathrm{\infty }$, that admits a $\mathcal{U}$-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 $\mathcal{U}$-frequently hypercyclic vectors, and to solve a question posed by Bès and Menet in 2015 on the existence of common $\mathcal{U}$-frequently hypercyclic subspaces.