Absolute continuity and hyponormal operators

Let T be a completely hyponormal operator, with the rectangular representation T = A + i B , on a separable Hilbert space. If 0 is not an eigenvalue of T * then T also has a polar factorization T = U P , with U unitary. It is known that A , B and U are all absolutely continuous operators. Conversely, given an arbitrary absolutely continuous selfadjoint A or unitary U , it is shown that there exists a corresponding completely hyponormal operator as above. It is then shown that these ideas can be used to establish certain known absolute continuity properties of various unitary operators by an appeal to a lemma in which, in one interpretation, a given unitary operator is regarded as a polar factor of some completely hyponormal operator. The unitary operators in question are chosen from a number of sources: the F. and M. Riesz theorem, dissipative and certain mixing transformations in ergodic theory, unitary dilation theory, and minimal normal extensions of subnormal contractions.