Weak cluster points of maximizing sequences on Banach spaces satisfying $\boldsymbol{(M_p)}$

TL;DR

This paper investigates the behavior of weak cluster points of maximizing sequences for bounded linear operators on Banach spaces with property (Mp), revealing a dichotomy in their extremal norms.

Contribution

It establishes a novel dichotomy result for the extremal norms of weak cluster points of maximizing sequences on Banach spaces with property (Mp).

Findings

Weak cluster points' norms are either 0 or 1.

Three classes of operators are characterized by this dichotomy.

The results connect norm-attainment and essential norm properties.

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.