On the Weak Maximizing Properties

TL;DR

This paper introduces a generalized weak maximizing property for norm-attaining operators, extending previous concepts by considering different topologies and providing new conditions for pairs of Banach spaces to possess this property.

Contribution

It defines a generalized version of the weak maximizing property using various topologies and establishes new sufficient conditions based on asymptotic uniform smoothness and convexity.

Findings

The pair (lex_p,lex_q) has the WMP.

New conditions imply many pairs have the maximizing property.

Extension of WMP to different topologies broadens its applicability.

Abstract

Quite recently, a new property related to norm-attaining operators has been introduced: the weak maximizing property (WMP). In this note, we define a generalised version of it considering other topologies than the weak one (mainly the weak$^*$ topology). We provide new sufficient conditions, based on the moduli of asymptotic uniform smoothness and convexity, which imply that a pair $(X,Y)$ enjoys a certain maximizing property. This approach not only allows us to (re)obtain as a direct consequence that the pair $(\ell_p,\ell_q)$ has the WMP, but also provides many more natural examples of pairs having a given maximizing property.