Equivalence classes of linear mappings on \(\mathcal B(\mathcal M)\) (Q432520)

Let \({\mathcal B}({\mathcal M})\) the \(C^*\)-algebra of all adjointable operators on a Hilbert \(C^*\)-module \({\mathcal M}\) over a \(C^*\)-algebra \({\mathcal A}\), and \({\mathcal L}({\mathcal B}({\mathcal M}))\) be the algebra of all linear operators on \({\mathcal B}({\mathcal M})\). The authors define and study on \({\mathcal L}({\mathcal B}({\mathcal M}))\) certain equivalence relations coming from different classes and properties of adjointable operators on \({\mathcal B}({\mathcal M})\) such as \({\mathcal A}\)-Fredholmness, semi-\({\mathcal A}\)-Fredholmness, compactness, and generalized invertibility.