The invariant subspace problem is the long-standing question whether every operator on a Hilbert space of dimension greater than one has a non-trivial invariant subspace. Although the problem is unsolved in the Hilbert space case, there are counter-examples for operators acting on certain well-known non-reflexive Banach spaces. These counter-examples are constructed by considering a single orbit and then extending continuously to a hounded linear map on the entire space. Based on this process, we introduce an operator which has properties closely linked with an orbit. We call this operator the orbit operator.
In the first part of the thesis, examples and basic properties of the orbit operator are discussed. Next, properties linking invariant subspaces to properties of the orbit operator are presented. Topics include the kernel and range of the orbit operator, compact operators, dilation theory, and Rotas theorem. Finally, we extend results obtained for strict contractions to contractions.
| Identifer | oai:union.ndltd.org:uvic.ca/oai:dspace.library.uvic.ca:1828/2089 |
| Date | 21 January 2010 |
| Creators | Deeley, Robin |
| Contributors | Sourour, A. R. |
| Source Sets | University of Victoria |
| Language | English, English |
| Detected Language | English |
| Type | Thesis |
| Rights | Available to the World Wide Web |
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