Let T be a contraction and A the strong limit of {T∗nTn}n≥1. We prove the following theorem: if a hyponormal contraction T does not have a nontrivial invariant subspace, then T is either a proper contraction of class 𝒞00 or a nonstrict proper contraction of class 𝒞10 for which A is a completely nonprojective nonstrict proper contraction. Moreover,
its self-commutator [T*,T] is a strict contraction.