On a preorder relation for contractions

An order relation for contractions on a Hilbert space can be introduced by stating that $A\preccurlyeq B$ if and only $A$ is unitarily equivalent to the restriction of $B$ to an invariant subspace. We discuss the equivalence classes associated to this relation, and identify cases in which they coincide with classes of unitary equivalence. The results extend those for completely nonunitary partial isometries obtained by Garcia, Martin, and Ross.