Smooth points in spaces of operators

For Banach spaces X, Y, in the space of bounded linear operators L(X,Y), we examine the relation between T∈L(X,Y) being a smooth point versus T⁎∈L(Y⁎,X⁎) being a smooth point. Motivated by some results in the recent paper of Paul et al., we give some sufficient conditions for the validity of such a statement for spaces of operators.