DOI: https://doi.org/10.15368/theses.2022.34

Available at: https://digitalcommons.calpoly.edu/theses/2467

#### 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

We will explore the numerical range of the block Toeplitz operator with symbol function \(\phi(z)=A_0+zA_1\), where \(A_0, A_1 \in M_2(\mathbb{C})\). A full characterization of the numerical range of this operator proves to be quite difficult and so we will focus on characterizing the boundary of the related set, \(\{W(A_0+zA_1) : z \in \partial \mathbb{D}\}\), in a specific case. We will use the theory of envelopes to explore what the boundary looks like and we will use geometric arguments to explore the number of flat portions on the boundary. We will then make a conjecture as to the number of flat portions on the boundary of the numerical range for any \(2 \times 2\) matrices \(A_0\) and \(A_1\). We finish by providing examples of flat portions on the boundary of the numerical range when \(A_0, A_1 \in M_n(\mathbb{C})\), for \(3 \leq n \leq 5\).