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 for every bounded linear operator $T$ from $E$ into $F$, the existence of a non-weakly null maximizing sequence for $T$ implies that $T$ attains its norm. This property was recently introduced in an article by R. Aron, D. Garc\'ia, 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 behaviour of 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 $p>2$ or $q<2$. We also show that, under certain conditions on $E$, the assumption that $(E, F)$ has the WMP for every Banach space $F$ implies that $E$ must be finite dimensional. On the other hand, we show that $(E, F)$ has the WMP for every reflexive space $E$ if and only if $F$ has the Schur property. We also give a complete characterization for the pairs $(\ell_s \oplus_p \ell_p, \ell_s \oplus_q \ell_q)$ to have the WMP by calculating the moduli of asymptotic uniform convexity of $\ell_s \oplus_p \ell_p$ and of asymptotic uniform smoothness of $\ell_s \oplus_q \ell_q$ when $1 < p \leq s \leq q < \infty$. We conclude the paper by discussing some variants of the WMP and presenting a list of open problems on the topic of the paper.