A note on generalized commuting contractions, and isometries

We investigate pairs of contractions [Formula: see text] with a unitary operator [Formula: see text] satisfying the commutation relations [Formula: see text] We extend a Hardy-type representation of Ando’s dilation for pure pairs of contractions satisfying the above commutation relations, recovering as a special case (see Theorem 3.1 of Ref. 6). Furthermore, we generalize the representation of BCL in Hardy space 11 to pairs of isometries [Formula: see text] that satisfy the commutation relations [Formula: see text] Finally, we present some immediate applications of our results.