Weak cluster points of maximizing sequences on Banach spaces satisfying $({\mathbf{M}}_{\mathbf{p}})$

Abstract

Given a bounded linear operator $T$ on a separable Banach space with property $(M_p)$, we prove that the smallest and the largest norm of weak cluster points of all maximizing sequences for $T$ can only take the values $0$ or $1$. The three classes of bounded linear operators emerging from the dichotomy of these extremal norm values coincides with the partition, created by considering the norm-attaining property and if the essential norm equals the norm.

6 pages, 1 figure, In the proof Proposition 3.2 (now Proposition 3.4), it was claimed MV is finite dimensional. This is unclear, but it is compact. I was informed about an article by Sreejith Siju, which has been added and given credit where credit is due (e.g. in the proof of the main theorem)