Absolute continuity and hyponormal operators
Let T be a completely hyponormal operator, with the rectangular representation T=A+iB, on a separable Hilbert space. If 0 is not an eigenvalue of T* then T also has a polar factorization T=UP, with U unitary. It is known that A,B and U are all absolutely continuous operators. Conversely, given an ar...
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Wiley
1981-01-01
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| Series: | International Journal of Mathematics and Mathematical Sciences |
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| Online Access: | http://dx.doi.org/10.1155/S0161171281000197 |
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| author | C. R. Putnam |
| author_facet | C. R. Putnam |
| author_sort | C. R. Putnam |
| collection | DOAJ |
| description | Let T be a completely hyponormal operator, with the rectangular representation T=A+iB, on a separable Hilbert space. If 0 is not an eigenvalue of T* then T also has a polar factorization T=UP, 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. |
| format | Article |
| id | doaj-art-770aaca348c844b09c48e093d08653e9 |
| institution | OA Journals |
| issn | 0161-1712 1687-0425 |
| language | English |
| publishDate | 1981-01-01 |
| publisher | Wiley |
| record_format | Article |
| series | International Journal of Mathematics and Mathematical Sciences |
| spelling | doaj-art-770aaca348c844b09c48e093d08653e92025-08-20T02:07:35ZengWileyInternational Journal of Mathematics and Mathematical Sciences0161-17121687-04251981-01-014232133510.1155/S0161171281000197Absolute continuity and hyponormal operatorsC. R. Putnam0Department of Mathematics, Purdue University, West Lafayette, Indiana 47907, USALet T be a completely hyponormal operator, with the rectangular representation T=A+iB, on a separable Hilbert space. If 0 is not an eigenvalue of T* then T also has a polar factorization T=UP, 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.http://dx.doi.org/10.1155/S0161171281000197selfadjoint operatorsunitary operatorshyponormal operatorsergodic theoryunitary dilationssubnormal operators. |
| spellingShingle | C. R. Putnam Absolute continuity and hyponormal operators International Journal of Mathematics and Mathematical Sciences selfadjoint operators unitary operators hyponormal operators ergodic theory unitary dilations subnormal operators. |
| title | Absolute continuity and hyponormal operators |
| title_full | Absolute continuity and hyponormal operators |
| title_fullStr | Absolute continuity and hyponormal operators |
| title_full_unstemmed | Absolute continuity and hyponormal operators |
| title_short | Absolute continuity and hyponormal operators |
| title_sort | absolute continuity and hyponormal operators |
| topic | selfadjoint operators unitary operators hyponormal operators ergodic theory unitary dilations subnormal operators. |
| url | http://dx.doi.org/10.1155/S0161171281000197 |
| work_keys_str_mv | AT crputnam absolutecontinuityandhyponormaloperators |