Let B (H) denote the algebra of operators on a complex Hilbert space H, and let u denote the class of operators A ∈ B(H) which satisfy the absolute value condition
. It is proved that if A ∈ u is a contraction, then either A has a nontrivial invariant subspace or A is a proper contraction and the nonnegative operator D =
2 is strongly stable. A Putnam-Fuglede type commutativity theorem is proved for contractions A in U, and it is shown that if normal subspaces of A ∈ U are reducing, then every compact operator in the intersection of the weak closure of the range of the derivation δA(X) = AX - XA with the commutant of A* is quasinilpotent.