Banach spaces with a unique nontrivial decomposition

Motivated by a problem of P. Koszmider we introduce the class of quasi-prime Banach spaces. This class lies between the classes of prime and primary Banach spaces. It is shown that for every $1<p<\infty$ there exists a strictly quasi-prime separable reflexive Banach space $\mathfrak{X}_p$ such that $\ell_p$ is a complemented subspace of $\mathfrak{X}_p$. A similar result also holds for the case of $\ell_1$ and $c_0$. More generally, for every separable decomposable prime Banach space $Y$ not containing $\ell_1$ there exists a strictly quasi-prime $\mathfrak{X}_Y$ containing $Y$ as a complemented subspace. We also investigate the operators acting on these spaces as well as the complemented subspaces of their finite powers.