On the weak maximizing properties

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 ( p, q ) has the WMP, but also provides many more natural examples of pairs having a given maximizing property.

Some remarks on the weak maximizing property

“…As a counterpart for the already mentioned paper [4], where some conditions for a pair of Banach spaces to have the WMP are given, the next theorem provides natural pairs of Banach spaces failing such a property. By a trivial Banach space we refer to a Banach space consisting of just the vector zero.…”

“…Once this result is established, Theorem 2.10 is proved automatically as a consequence of [4,Theorem 3.1]. In order to prove Theorem 2.11, we need the following two straightforward lemmas.…”

“…Let us finish this section by discussing some variants of the WMP. In [4], the authors considered the bidual-WMP, that is, a pair (E, F ) of Banach spaces has the bidual-WMP if for every operator T ∈ L(E, F ), the existence of a non-weakly null maximizing sequence for T implies that T * * ∈ NA(E * * , F * * ) (see [4,Definition 5.13]). This seems to be natural to be considered due to Lindenstrauss result [10, Theorem 1], where he proved that the set of all operators in which their second adjoints are norm-attaining is dense in L(E, F ).…”

“…Even more recent is the paper [4], where the authors have provided a new plan of action to deal with the WMP (and its variants) by using the modulus of asymptotic uniform convexity and the modulus of asymptotic uniform smoothness (see [4,Theorem 3.1]). In particular, using these tools they manage to cover all the positive known examples from [2] and, furthermore, to present a big array of new pairs of Banach spaces satisfying the WMP.…”

“…On the other hand, it is immediate that every operator from a reflexive Banach space into a Banach space with the Schur property is compact and, therefore, attains its norm. Thus, if F has the Schur property, then (E, F ) has the WMP for every reflexive Banach space E (see [4,Proposition 5.2]). In Theorem 2.6 it is shown that, indeed, the Schur property is characterized by this property, i.e.…”