[Author] Oleg Reinov [Title] On linear operators with s-nuclear adjoints, $0< s\le 1$ [AMS Subj-class] 47B10 Operators belonging to operator ideals [Abstract] We prove that 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 $s,$ 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. [Comments] LaTeX, English, 13 pp. [Contact e-mail] orein51@mail.ru