Some Equivalence Classes of Operators on B(H) (original) (raw)

Let L(B(H)) be the algebra of all linear operators on B(H) and P be a property on B(H). For φ1, φ2 ∈ L(B(H)), we say that φ1∼ P φ2, whenever φ1(T ) has property P, if and only if φ2(T ) has this property. In particular, if I is the identity map on B(H), then φ∼ P I means that φ preserves property P in both directions. Each property P produces an equivalence relation on L(B(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.