It is shown that if a paranormal contraction T has no nontrivial invariant subspace, then it is a proper contraction. Moreover, the nonnegative operator Q = T^2*T² - 2T^*T ＋ I also is a proper contraction. If a quasihyponormal contraction has no nontrivial invariant subspace then, in addition, its defect operator D is a proper contraction and its itself-commutator is a trace-class strict contraction. Furthermore, if one of Q or D is compact, then so is the other, and Q and D are strict ontraction.