Approximation properties $A{P}_s$ and p-nuclear operators (the case where $0<s\le 1$)

Among other things, it is shown that there exist Banach spaces $Z$ and $W$ such that $Z^{\ast \ast }$ and $W$ have bases, and for every $p\in [1,2)$ there is an operator $T:W\to Z$ that is not $p$-nuclear but $T^{\ast \ast }$ is $p$-nuclear.