SOME EQUIVALENCE CLASSES OF OPERATORS ON B(H)

Let L(B(H)) be the algebra of all linear operators on B(H) and P be a property on B(H). For phi(1),phi(2) is an element of L(B(H)), we say that phi(1) phi(P) phi(2), whenever phi(1)(T) has property P, if and only if phi(2) (T) has this property. In particular, if I is the identity map on B(H), then phi similar to(P) I means that phi 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.