Weak cluster points of maximizing sequences on Banach spaces satisfying $\boldsymbol{(M_p)}$
Functional Analysis
2026-02-25 v2
Abstract
Given a bounded linear operator on a separable Banach space with property , we prove that the smallest and the largest norm of weak cluster points of all maximizing sequences for can only take the values or . 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.
Keywords
Cite
@article{arxiv.2508.11420,
title = {Weak cluster points of maximizing sequences on Banach spaces satisfying $\boldsymbol{(M_p)}$},
author = {David Norrbo},
journal= {arXiv preprint arXiv:2508.11420},
year = {2026}
}
Comments
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)