Available at: https://digitalcommons.calpoly.edu/theses/2492

#### Date of Award

6-2022

#### Degree Name

MS in Mathematics

#### Department/Program

Mathematics

#### College

College of Science and Mathematics

#### Advisor

Linda Patton

#### Advisor Department

Mathematics

#### Advisor College

College of Science and Mathematics

#### Abstract

One of the many characterizations of compact operators is as linear operators which

can be closely approximated by bounded finite rank operators (theorem 25). It is

well known that the numerical range of a bounded operator on a finite dimensional

Hilbert space is closed (theorem 54). In this thesis we explore how close to being

closed the numerical range of a compact operator is (theorem 56). We also describe

how limited the difference between the closure and the numerical range of a compact

operator can be (theorem 58). To aid in our exploration of the numerical range of

a compact operator we spend some time examining its spectra, as the spectrum of a

bounded operator is closely tied to its numerical range (theorem 45). Throughout,

we use the forward shift operator and the diagonal operator (example 1) to illustrate

the exceptional behavior of compact operators.

*Proof of Payment*