Let E be a Banach space and
B(E) be the space of all bounded linear operators on E. It was shown by Schatten,
that if E is a conjugate space then B(E) is isometrically isomorphic to a conjugate
space. The fact that for an arbitrary Banach space, the unit ball of B(E) has extreme
points suggests that B(E) might always be a conjugate space. In this paper it is
proved that if E has an unconditional basis and is not isomorphic to a conjugate
space, then B(E) is not isomorphic to a conjugate space. An even stronger result is
proved.
Furthermore, it is shown that if E has an unconditional basis or a complemented
subspace with an unconditional basis, then the space of all compact linear operators
on E is not isomorphic to a conjugate space.
