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\\).
","abstract_html":"<p>We will explore the numerical range of the block Toeplitz operator with symbol function \\phi(z)=A0+zA1, where A0, A1 \\in M2(\\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(A0+zA1) : 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 A0 and A1. We finish by providing examples of flat portions on the boundary of the numerical range when A0, A1 \\in Mn(\\mathbb{C}), for \\(3 \\leq n \\leq 5\\).</p>","abstract_has_math":true,"creators":["Randell, Brooke"],"institution":null,"degree_name":"MS in Mathematics","degree_level":null,"degree_discipline":"Mathematics","degree_department":null,"school":null,"contributors":["Linda Patton","Mathematics","College of Science and Mathematics"],"advisors":[],"committee_chairs":[],"committee_members":[],"year":2022,"date_issued":"2022-06-01T07:00:00Z","date_published":"2022-06-01T07:00:00Z","updated_at":"2026-07-24T01:32:29Z","subjects":["Analysis","Other Mathematics"],"languages":[],"rights":[],"rights_urls":[],"identifier_entries":[{"key":"dc:identifier","label":"Identifier","values":["10.15368/theses.2022.34"],"render_values":[{"text":"10.15368/theses.2022.34","href":"https://doi.org/10.15368/theses.2022.34","code":true}]}]},"links":{"outbound_url":"https://digitalcommons.calpoly.edu/theses/2467","outbound_label":"Repository record","outbound_source":"dc:identifier"},"metadata_groups":[{"id":"people","label":"People","entries":[{"key":"dc:contributor","label":"Contributor","values":["Linda Patton","Mathematics","College of Science and Mathematics"]},{"key":"dc:creator","label":"Author","values":["Randell, Brooke"]}]},{"id":"academic_context","label":"Academic Context","entries":[{"key":"dc:date.available","label":"Dc Date Available","values":["2022-06-05T07:00:00Z"]},{"key":"thesis:degree_discipline","label":"Discipline","values":["Mathematics"]},{"key":"thesis:degree_name","label":"Degree Name","values":["MS in Mathematics"]}]},{"id":"subjects_keywords","label":"Subjects and Keywords","entries":[{"key":"dc:subject","label":"Dc Subject","values":["Analysis","Other Mathematics"]}]},{"id":"identifiers","label":"Identifiers","entries":[{"key":"dc:identifier","label":"Identifier","values":["https://digitalcommons.calpoly.edu/theses/2467","10.15368/theses.2022.34"]}]},{"id":"additional","label":"Additional Metadata","entries":[{"key":"dc:description.abstract","label":"Abstract","values":["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\\).
"]},{"key":"dc:title","label":"Title","values":["Exploring the Numerical Range of Block Toeplitz Operators"]}]}],"canonical_facts":{"dc:contributor":["Linda Patton","Mathematics","College of Science and Mathematics"],"dc:creator":["Randell, Brooke"],"dc:date.available":["2022-06-05T07:00:00Z"],"dc:description.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\\).
"],"dc:identifier":["https://digitalcommons.calpoly.edu/theses/2467","10.15368/theses.2022.34"],"dc:subject":["Analysis","Other Mathematics"],"dc:title":["Exploring the Numerical Range of Block Toeplitz Operators"],"thesis:degree_discipline":["Mathematics"],"thesis:degree_name":["MS in Mathematics"]},"updated_at":"2026-07-24T01:32:29Z"}