On linear operators with p-nuclear adjoints

Abstract

If $p\in [1,+\infty ]$ 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 $l_{p^{\prime }}\to l_1.$ On the other hand, there is a Banach space $W$ such that $W^{\ast \ast }$ has a basis and such that for each $p\in [1,+\infty ],p\ne 2,$ 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 $W.$

6 pages, AMSTeX