On linear operators with s-nuclear adjoints, 0<s⩽1

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 Z0 and a non-nuclear operator T:Z0⁎⁎→Z0 so that Z0⁎⁎ has a Schauder basis, Z0⁎⁎⁎ has the APs for every s∈(0,r) and T⁎ is r-nuclear.