Some remarks on the weak maximizing property

A pair of Banach spaces (E,F) is said to have the weak maximizing property (WMP, for short) if every bounded linear operator T from E into F, existence a non-weakly null sequence implies that attains its norm. This was recently introduced in paper by R. Aron, D. García, Pelegrino and E. Teixeira, raising several open questions. The aim present contribute better knowledge WMP limitations. Namely, we provide sufficient conditions fail study behavior this with respect quotients, subspaces, direct sums, which gate consequences. For instance, deal pairs form (Lp[0,1],Lq[0,1]), proving these whenever p>2 or q<2. We also show that, under certain on E, assumption has space F must be finite dimensional. On other hand, reflexive only Schur property. give complete characterization (ℓs⊕pℓp,ℓs⊕qℓq) calculating moduli asymptotic uniform convexity ℓs⊕pℓp smoothness ℓs⊕qℓq when 1<p⩽s⩽q<∞. conclude discussing some variants presenting list problems topic paper.