2026-07-072026-07-07http://salesiana.dossiersoluciones.com/handle/123456789/76345Let $\CR$ be the set of all bounded linear operators between Hilbert spaces $\cH, \cK$. This paper is devoted to the study of the topological properties of $\CR$ if certain natural metrics are considered on it. We also define an action of the group $\G_\cH\times\G_\cK$ on $\CR$ and determine the orbits of this action. These orbits determine a stratification of the set of Fredholm and semi-Fredholm operators. Finally, we calculate the distance, with respect to some of the metrics mentioned above, between different orbits of $\CR$.Functional Analysis46C07; 47A62; 46C05text