  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.
