ON THE EXISTENCE OF NON-NORM-ATTAINING OPERATORS

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, 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 that does not attain its norm. This allows us to provide the following generalisation 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 (pointwise-)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) $(\mathcal {L}(E,F), \tau _c)^* = (\mathcal {L}(E, F), \left \Vert \cdot \right \Vert )^*$ , where $\tau _c$ denotes the topology of compact convergence. We conclude the article by presenting a characterisation of the Schur property in terms of norm-attaining operators.

The norm attainment problem for functions of projections

The paper is concerned with the problem of identifying the norm attaining operators in the von Neumann algebra generated by two orthogonal projections on a Hilbert space. Every skew projection on that Hilbert space is contained in such an algebra and hence the results of the paper also describe functions of skew projections and their adjoints that attain the norm.