On linear operators with s-nuclear adjoints: $0< s \le 1$

If $s\in (0,1]$ and $T$ is a linear operator with $s$-nuclear adjoint from a Banach space $X$ to a Banach space $Y$ and if one of the spaces $X^*$ or $Y^{***}$ has the approximation property of order $s,$ $AP_s,$ then the operator $T$ is nuclear. The result is in a sense exact. For example, it is shown that for each $r\in (2/3, 1]$ there exist a Banach space $Z_0$ and a non-nuclear operator $T: Z_0^{**}\to Z_0$ so that $Z_0^{**}$ has a Schauder basis, $Z_0^{***}$ has the $AP_s$ for every $s\in (0,r)$ and $T^*$ is $r$-nuclear.