  In this paper we provide necessary and sufficient conditions for the
existence of non-norm-attaining operators in $\mathcal{L}(E, F)$. By using a
theorem due to Pfitzner on James boundaries, we show that if there exists a
relatively compact set $K$ of $\mathcal{L}(E, F)$ (in the weak operator
topology) such that $0$ is an element of its closure (in the weak operator
topology) but it is not in its norm closed convex hull, then we can guarantee
the existence of an operator which does not attain its norm. This allows us to
provide the following generalization of results due to Holub and Mujica. If $E$
is a reflexive space, $F$ is an arbitrary Banach space, and the pair $(E, F)$
has the bounded compact approximation property, then the following are
equivalent:
  (i) $\mathcal{K}(E, F) = \mathcal{L}(E, F)$;
  (ii) Every operator from $E$ into $F$ attains its norm;
  (iii) $(\mathcal{L}(E,F), \tau_c)^* = (\mathcal{L}(E, F), \| \cdot \|)^*$;
  where $\tau_c$ denotes the topology of compact convergence. We conclude the
paper presenting a characterization of the Schur property in terms of
norm-attaining operators.
