Pointwise products of some Banach function spaces and factorization

The well-known factorization theorem of Lozanovski{\u \i} may be written in the form $L^{1}\equiv E\odot E^{\prime}$, where $\odot$ means the pointwise product of Banach ideal spaces. A natural generalization of this problem would be the question when one can factorize $F$ through $E$, i.e., when $F\equiv E\odot M(E, F) \,$, where $M(E, F)$ is the space of pointwise multipliers from $E$ to $F$. Properties of $M(E, F)$ were investigated in our earlier paper [KLM12] and here we collect and prove some properties of the construction $E\odot F$. The formulas for pointwise product of Calder\'{o}n-Lozanovski{\u \i} $E_{\varphi}$ spaces, Lorentz spaces and Marcinkiewicz spaces are proved. These results are then used to prove factorization theorems for these spaces. Finally, it is proved in Theorem 11 that under some natural assumptions, a rearrangement invariant Banach function space may be factorized through Marcinkiewicz space.