Some equivalence classes of operators on B(H)

Let $\mathcal L(\mathcal B(\mathcal H))$ be the algebra of all linear operators on‎ ‎$\mathcal B(\mathcal H)$ and $\mathcal P$ be a property on‎ ‎$\mathcal B(\mathcal H)$‎. ‎For $\phi_{1},\phi_{2}\in \mathcal‎ ‎L(\mathcal B(\mathcal H))$‎, ‎we say that‎ ‎$\phi_{1}{\sim}_{_{\mathcal P}} \phi_{2}$‎, ‎whenever $\phi_{1}(T) $‎ ‎has property $\mathcal P$‎, ‎if and only if $\phi_{2}(T)$ has this‎ ‎property‎. ‎In particular‎, ‎if $\mathcal I$ is the identity map on‎ ‎$\mathcal B(\mathcal H)$‎, ‎then $\phi{\sim}_{_{\mathcal P}}‎ ‎\mathcal I$ means that $\phi$ preserves property $\mathcal P$ in‎ ‎both directions‎. ‎Each property $\mathcal P$ produces an‎ ‎equivalence relation on $\mathcal L(\mathcal B(\mathcal H))$‎. ‎We‎ ‎study the relation between equivalence classes with respect to‎ ‎different properties such as being Fredholm‎, ‎semi-Fredholm‎, ‎compact‎, ‎finite rank‎, ‎generalized invertible‎, ‎or having a‎ ‎specific semi-index‎.