Absolute continuity and hyponormal operators. (1981) - Putnam, C. R.
Abstract : LetT be a completely hyponormal operator, with the rectangular representationT = A + i B, on a separable Hilbert space. If0 is not an eigenvalue ofT * thenT also has a polar factorizationT = U P, withU unitary. It is known thatA, B andU are all absolutely continuous operators. Conversely, given an arbitrary absolutely continuous selfadjointA or unitaryU, 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.