Some remarks on the weak maximizing property

[摘要] A pair of Banach spaces (E, F) is said to have the weak maximizing property (WMP, for short) if for every bounded linear operator Tfrom Einto F, the existence of a non-weakly null maximizing sequence for Timplies that Tattains its norm. This property was recently introduced in a paper by R. Aron, D. Garcia, D. Pelegrino and E. Teixeira, raising several open questions. The aim of the present paper is to contribute to the better knowledge of the WMP and its limitations. Namely, we provide sufficient conditions for a pair of Banach spaces to fail the WMP and study the behaviorof this property with respect to quotients, subspaces, and direct sums, which open the gate to present several consequences. For instance, we deal with pairs of the form (L-p[0, 1], L-q[0, 1]), proving that these pairs fail the WMP whenever p > 2or q< 2. We also show that, under certain conditions on E, the assumption that (E, F) has the WMP for every Banach space Fimplies that Emust be finite dimensional. On the other hand, we show that (E, F) has the WMP for every reflexive space Eif and only if Fhas the Schur property. We also give a complete characterization for the pairs (l(s) circle plus(p) l(p), l(s) circle plus(q) l(q)) to have the WMP by calculating the moduli of asymptotic uniform convexity of l(s) circle plus(p)l(p) and of asymptotic uniform smoothness of l(s) circle plus(q)l(q) when 1 < p <= s <= q< infinity. We conclude the paper by discussing some variants of the WMP and presenting a list of open problems on the topic of the paper. (c) 2021 Elsevier Inc. All rights reserved.

[发布日期] 2021-12-15

[关键词] Maximizing sequence;Norm-attaining operators;James theorem