Hereditarily hypercyclic subspaces

We say that a sequence of operators $(T_n)$ possesses hereditarily hypercyclic subspaces along a sequence $(n_k)$ if for any subsequence $(m_k)\subset(n_k)$, the sequence $(T_{m_k})$ possesses a hypercyclic subspace. While so far no characterization of the existence of hypercyclic subspaces in the case of Fr\'{e}chet spaces is known, we succeed to obtain a characterization of sequences $(T_n)$ possessing hereditarily hypercyclic subspaces along $(n_k)$, under the assumption that the sequence $(T_n)$ satisfies the Hypercyclicity Criterion along $(n_k)$. We also obtain a characterization of operators possessing a hypercyclic subspace under the assumption that $T$ satisfies the Frequent Hypercyclicity Criterion.