On linear operators with p-nuclear adjoints

If $p\in [1,+\mathrm{\infty }]$ and $T$ is a linear operator with $p$-nuclear adjoint from a Banach space $X$ to a Banach space $Y$ then if one of the spaces $X^{\ast }$ or $Y^{\ast \ast \ast }$ has the approximation property, then $T$ belongs to the ideal $N^p$ of operators which can be factored through diagonal oparators $l_{p’}\to l_1.$Ontheotherhand,thereisaBanachspace$W$suchthat$W^{**}$ has a basis and such that for each $p\in [1,+\mathrm{\infty }],p\mathrm{\ne }2,$ there exists an operator $T:W^{\ast \ast }\to W$ with $p$-nuclear adjoint that is not in the ideal $N^p,$ as an operator from $W^{\ast \ast }$ to $W\mathrm{.}$