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.

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