A Banach space $X$ is said to be prime if every infinite dimensional complemented subspace is isomorphic to the space $X$. The space $X$ is primary if it has an infinite dimensional subspace $Y$ such that every complemented subspace is either isomorphic to $X$ or to the subspace $Y$. The space $X$ is quasiprime if it is primary and the only decomposition of $X$, as a direct sum, into two infinite dimensional subspaces is $X +Y$. Prime spaces are quasiprime which are primary. Also there are examples that separate the above classes.
A Banach space $X$ is hereditarily prime (primary, quasiprime) if every infinite dimensional subspace is prime (primary, quasiprime). Hilbert spaces are hereditarily prime. Also there exists a non-Hilbertian space $X$ which is hereditarily quasiprime.
Question I Is every hereditarily prime space isomorphic to a Hilbert space?
Question II Do there exist subspaces of $\ell_p$, which are not primary?