On the Numerical Range of Compact Operators

<p>One of the many characterizations of compact operators is as linear operators which<br />can be closely approximated by bounded finite rank operators (theorem 25). It is<br />well known that the numerical range of a bounded operator on a finite dimensional<br />Hilbert space is closed (theorem 54). In this thesis we explore how close to being<br />closed the numerical range of a compact operator is (theorem 56). We also describe<br />how limited the difference between the closure and the numerical range of a compact<br />operator can be (theorem 58). To aid in our exploration of the numerical range of<br />a compact operator we spend some time examining its spectra, as the spectrum of a<br />bounded operator is closely tied to its numerical range (theorem 45). Throughout,<br />we use the forward shift operator and the diagonal operator (example 1) to illustrate<br />the exceptional behavior of compact operators.</p>

Name thesis:degree_name

MS in Mathematics

Discipline thesis:degree_discipline

Mathematics

Year dc:date.available

2022

Author dc:creator

• Dabkowski, Montserrat

Contributors dc:contributor

• Linda Patton
• Mathematics
• College of Science and Mathematics

• Numerical Range
• Compact Operator
• Spectra
• Eigenvalue
• Analysis
• Other Mathematics

Identifier

10.15368/theses.2022.54

OAI identifier oai:identifier

oai:digitalcommons.calpoly.edu:theses-4066