Abstract. For non-reflexive Banach spaces $X,Y$, for a very smooth point in the space of compact linear operators ${\mathcal K}(X,Y)$, we give several sufficient conditions for the adjoint to be a very smooth point in ${\mathcal K}(Y^\ast , X^\ast )$. We exhibit a new class of extreme points in the dual unit ball of injective product spaces. These ideas are also related to Birkhoff--James orthogonality in spaces of operators.

DOI: 10.14232/actasm-018-809-2

AMS Subject Classification (1991): 47L05, 46B28, 46B25

Keyword(s): smooth points, very smooth points, adjoints of operators, spaces of operators, essential norm, injective and projective tensor product spaces

Received June 23, 2018 and in final form January 21, 2019. (Registered under 59/2018.)