International Journal of Mathematics and Mathematical Sciences
Volume 4 (1981), Issue 2, Pages 321-335
Absolute continuity and hyponormal operators
Department of Mathematics, Purdue University, West Lafayette 47907, Indiana, USA
Received 18 July 1980
Copyright © 1981 C. R. Putnam. This is an open access article distributed under the Creative Commons Attribution License, which permits unrestricted use, distribution, and reproduction in any medium, provided the original work is properly cited.
Let be a completely hyponormal operator, with the rectangular representation , on a separable Hilbert space. If is not an eigenvalue of then also has a polar factorization , with unitary. It is known that and are all absolutely continuous operators. Conversely, given an arbitrary absolutely continuous selfadjoint or unitary , 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.