One of the many characterizations of compact operators is as linear operators whichcan be closely approximated by bounded finite rank operators (theorem 25). It iswell known that the numerical range of a bounded operator on a finite dimensionalHilbert space is closed (theorem 54). In this thesis we explore how close to beingclosed the numerical range of a compact operator is (theorem 56). We also describehow limited the difference between the closure and the numerical range of a compactoperator can be (theorem 58). To aid in our exploration of the numerical range ofa compact operator we spend some time examining its spectra, as the spectrum of abounded operator is closely tied to its numerical range (theorem 45). Throughout,we use the forward shift operator and the diagonal operator (example 1) to illustratethe exceptional behavior of compact operators.
| Identifer | oai:union.ndltd.org:CALPOLY/oai:digitalcommons.calpoly.edu:theses-4066 |
| Date | 01 June 2022 |
| Creators | Dabkowski, Montserrat |
| Publisher | DigitalCommons@CalPoly |
| Source Sets | California Polytechnic State University |
| Detected Language | English |
| Type | text |
| Format | application/pdf |
| Source | Master's Theses |
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