Volume 35 • Number 3 • 2012   • Equivalence Classes of Linear Mappings on $\mathcal B(\mathcal M)$ S. Hejazian and T. Aghasizadeh Abstract. Let $\mathcal M$ be a Hilbert $C^*$-module over the $C^*$-algebra $\mathcal A$, $\mathcal B(\mathcal M)$ the $C^*$-algebra of all adjointable operators on $\mathcal M$, $\mathcal L(\mathcal B(\mathcal M))$ the algebra of all linear operators on $\mathcal B(\mathcal M)$. For a property $\mathcal P$ on $\mathcal B(\mathcal M)$ and $\phi_{1},\phi_{2}\in \mathcal L(\mathcal B(\mathcal M))$ we say that $\phi_{1}{\sim}_{_{\mathcal P}} \phi_{2}$, whenever for every $T\in\mathcal B(\mathcal M)$, $\phi_{1}(T) $ has property $\mathcal P$ if and only if $\phi_{2}(T)$ has this property. Each property $\mathcal P$ produces an equivalence relation on $\mathcal L(\mathcal B(\mathcal M))$. If $\mathcal I$ denotes the identity map on $\mathcal B(\mathcal M)$ it is clear that $\phi{\sim}_{_{\mathcal P}} \mathcal I$ means that $\phi$ preserves and reflects property $\mathcal P$. We are going to study the equivalence classes with respect to different properties such as being $\mathcal A$-Fredholm, semi-$\mathcal A$-Fredholm, compact and generalized invertible. 2010 Mathematics Subject Classification: 47B48, 46L08, 47A53. Full text: PDF