The approximation property of order $(p,q)$ in Banach spaces

We introduce the approximation property of order ($p,q$) in Banach spaces (in short $AP_{pq}$) to study topological properties of the space $D_{q'p'}(E,F)$ of ($q',p'$)-dominated operators between the Banach spaces $E$ and $F$. After some equivalent formulations of the $AP_{pq}$, we characterize the reflexivity of $D_{q'p'}(E,F)$ when $E$ has the $AP_{qp}$ or $F'$ has the $AP_{pq}$ and we give sufficient conditions for $E\hat\otimes_{\alpha_{pq}}F$ and $E\hat\otimes_{\alpha'_{pq}}F$ to be weakly sequentially complete.