ON THE EXISTENCE OF NON-NORM-ATTAINING OPERATORS

Abstract In this article, 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, show that if there exists relatively compact set K (in weak operator topology) such $0$ is an element its closure but it not norm-closed convex hull, then can guarantee does attain norm. This allows us following generalisation results Holub Mujica. If E reflexive space, F arbitrary Banach space pair $(E, has (pointwise-)bounded approximation property, are equivalent: (i) {K}(E, F) = \mathcal ; (ii) Every from into attains norm; (iii) $(\mathcal {L}(E,F), \tau _c)^* (\mathcal F), \left \Vert \cdot \right )^*$ , where $\tau _c$ denotes topology convergence. We conclude article by presenting characterisation Schur property terms norm-attaining operators.