The main goal of this article is to show that for every (reflexive) infinite-dimensional Banach space X there exists a reflexive Banach space Y and \(T, R \in \mathcal {L}(X,Y)\) such that R is a rank-one operator, \(\Vert T+R\Vert >\Vert T\Vert \) but \(T+R\) does not attain its norm. This answers a question posed by Dantas and the first two authors. Furthermore, motivated by the parallelism exhibited in the literature between the V-property introduced by Khatskevich, Ostrovskii and Shulman and the weak maximizing property introduced by Aron, García, Pellegrino and Teixeira, we also study the relationship between these two properties and norm-attaining perturbations of operators.
