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

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