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Date of Award

2014

Document Type

Restricted Thesis: Campus only access

Degree Name

Bachelor of Science

Department

Mathematics

First Advisor

Dr. William T. Ross

Abstract

In this thesis, we will explore Toeplitz matrices and their algebras. The Toeplitz matrices are square matrices with complex entries which are constant along their diagonals. They are known to be a linear subspace of all of the n×n matrices but they are not an algebra. We will explore several problems related to algebras one can form with Toeplitz matrices.

We begin this thesis with a general discussion of the relationship between the Toeplitz matrices and Toeplitz operators on the classical Hardy space. We will see that the Toeplitz matrices are compressions of Toeplitz operators to the space of polynomials of at most a certain fixed degree. Along the way, we will state some results of Coburn [3, 4] about algebras generated by certain Toeplitz operators to set the stage for related problems for Toeplitz matrices.

We then focus on the main theme of this thesis, the algebras within the Toeplitz matrices. These maximal algebras have been classified by Zimmer- man [12] and explored in greater generality by Sedlock [9]. We will review their work. The problem we explore here is: when can two seemingly dif- ferent maximal algebras of the Toeplitz matrices be viewed as “the same”? We will explore two version of “sameness”: spatial similarity and spatial isomorphism. Our techniques will connect to earlier work of Clark [2] on a certain class of unitary matrices related to the compressed shift.

We next focus on the following problem: Can every matrix be written as a product of Toeplitz matrices? The answer turns out to be yes and was recently proved by Ye and Lim [11]. We will review their techniques, which involve some algebraic geometry. We then proceed to bring in our earlier work on maximal algebras of Toeplitz matrices and show that the set of n × n matrices can be written as a finite product of maximal algebras of Toeplitz matrices.

At the end of this thesis, we state an open problem relating to products of Toeplitz operators on the Hardy space.

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