On Spectral Operators in Finite von Neumann Algebras
| dc.contributor.advisor | Dykema, Kenneth J | |
| dc.contributor.committeeMember | Anshelevich, Michael | |
| dc.contributor.committeeMember | Smith, Roger R | |
| dc.contributor.committeeMember | Pourahmadi, Mohsen | |
| dc.creator | Krishnaswamy-Usha, Amudhan | |
| dc.date.accessioned | 2021-05-11T22:08:48Z | |
| dc.date.available | 2022-12-01T08:18:10Z | |
| dc.date.created | 2020-12 | |
| dc.date.issued | 2020-12-04 | |
| dc.date.submitted | December 2020 | |
| dc.date.updated | 2021-05-11T22:08:49Z | |
| dc.description.abstract | An operator on a Hilbert space is said to be spectral if it has a suitably well-behaved `idempotent-valued' spectral measure. Dunford introduced these operators and also provided the following characterization: An operator is spectral iff it is similar to the sum of a normal operator and a quasinilpotent operator that commute with each other. Operators in a von Neumann algebra with a normal, faithful, tracial state have an associated spectral measure (called the Brown measure) and invariant projections (the Haagerup-Schultz projections), which behave well with respect to the Brown measure. In this paper, we study the angles between the Haagerup-Schultz projections for such operators. We show that an operator in a finite von Neumann algebra is similar to the sum of a normal operator and a commuting s.o.t.-quasinilpotent operator iff the angles between its Haagerup-Schultz projections are uniformly bounded away from zero. This lets us provide a new characterization of spectral operators in finite von Neumann algebras. We then estimate the angles between the Haagerup-Schultz projections for a class of operators from free probability called the DT-operators. These involve new estimates on the norms of algebra-valued circular operators. We then show, subject to some mild regularity conditions on the Brown measure of a DT-operator, that they fail to be spectral. This provides a large class of non-spectral but decomposable operators in a finite von Neumann algebra. | en |
| dc.format.mimetype | application/pdf | |
| dc.identifier.uri | https://hdl.handle.net/1969.1/193006 | |
| dc.language.iso | en | |
| dc.subject | operator algebras | en |
| dc.subject | finite von Neumann algebra | en |
| dc.subject | Haagerup-Schultz projection | en |
| dc.subject | spectral operators | en |
| dc.subject | circular operator | en |
| dc.subject | decomposable operators | en |
| dc.title | On Spectral Operators in Finite von Neumann Algebras | en |
| dc.type | Thesis | en |
| dc.type.material | text | en |
| local.embargo.terms | 2022-12-01 | |
| local.etdauthor.orcid | 0000-0002-8361-1426 | |
| thesis.degree.department | Mathematics | en |
| thesis.degree.discipline | Mathematics | en |
| thesis.degree.grantor | Texas A&M University | en |
| thesis.degree.level | Doctoral | en |
| thesis.degree.name | Doctor of Philosophy | en |