Given a bounded linear operator T on a separable Banach space with property (Mₚ), 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.
David Norrbo (Mon,) studied this question.