On linear operators with s-nuclear adjoints,

We prove that if s ∈ ( 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 ∈ ( 2 / 3 , 1 ] there exist a Banach space Z 0 and a non-nuclear operator T : Z 0 ⁎ ⁎ → Z 0 so that Z 0 ⁎ ⁎ has a Schauder basis, Z 0 ⁎ ⁎ ⁎ has the AP s for every s ∈ ( 0 , r ) and T ⁎ is r-nuclear.