Rank-one perturbations and norm-attaining operators

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, $\|T+R\|>\|T\|$ but $T+R$ does not attain its norm. This answers a question posed by S. Dantas and the first two authors. Furthermore, motivated by the parallelism exhibited in the literature between the $V$-property introduced by V.A. Khatskevich, M.I. Ostrovskii and V.S. Shulman and the weak maximizing property introduced by R.M. Aron, D. Garc\'ia, D. Pellegrino and E.V. Teixeira, we also study the relationship between these two properties and norm-attaining perturbations of operators.