Electronic Theses and Dissertations

Sana Kazemi

Date

2023

Document Type

Dissertation

Degree Name

Doctor of Philosophy

Department

Mathematical Sciences

Committee Chair

Fernanda Botelho

Committee Member

Pei-Kee Lin

Committee Member

Anna Kaminska

Committee Member

Bentuo Zheng

Committee Member

Mate Wierdl

Abstract

This dissertation contributes to the growing body of the literature relating isometric properties of sequence spaces and contains interesting mathematical concepts. Structured into four chapters, the dissertation begins with an introduction (Chapter 1) and provides definitions and background information on specific classes of operators in Chapter 2. Chapter 3 introduces the concept of algebraic reflexivity and explores conditions under which this property holds. Our research reveals that spaces with trivial Hermitian operators possess algebraic reflexivity, particularly within their isometry groups. Chapter 4 is dedicated to exploring the isometric properties of special classes of sequence spaces namely $X_n$, Schreier and Tsirelson-type space. Some results in this chapter and sections appeared in the reference (ref). In this chapter, we define $X_n$ space (ref). After establishing some geometric properties of $X_n$, it is shown that the isometry group of $X_n$ consists of standard isometries and is algebrically reflexive. Next, we introduce the Schreier space (ref), highlighting its similarities to $X_n$ in terms of geometric properties. We also confirm the presence of a diagonal standard group of isometries in the Schreier space, offering an alternative and concise proof using the form of extreme points, building upon the work of Antunes and Beanland (ref). Tsirelson space is the first example of a reflexive Banach space which has no isomorphic copy of $\ell _p$ nor $c_{0}$. Our investigation extends to Tsirelson-type space, which represents an intermediate space within the Tsirelson norm. By combining the infinite norm with the $n$-norm featured in $X_n$ space, we create this sequence space, denoted as $T[\theta, A_n]$. We explore various values of $\theta$ and $A_n$ to gain structural insights. Then we obtain two results, first one is about for $\theta \in (\frac{1}{3},1)$ the isometry group of $T[\theta, A_3]$ consist of standard isometries. Additionally, we establish that for $\theta \leq \frac{1}{n}$, $T[\theta, A_n]$ supports standard isometries. In the final section of our work, we delve into the structure of $T[\theta,\mathcal{F}]$, where $\mathcal{F}$ is any regular family, and $0 < \theta < 1$. The Tsirelson space $T[\theta,\mathcal{F}]$ represents the completion of $c_{00}$ with a specific norm, $\|\cdot\|_{\theta, \mathcal{F}}$, satisfying certain conditions (ref). We demonstrate that if $\mathcal{F}$ possesses property (s) and $U$ is an isometry from $T[\theta,\mathcal{F}]$ into itself, then specific properties hold regarding the permutation $\pi$ and the values $a_j$, offering a partial solution to an open question proposed by Antunes and Beanland (ref).

Comments

Data is provided by the student.

Library Comment

Dissertation or thesis originally submitted to ProQuest

Notes

Open Access

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