## Electronic Theses and Dissertations

## Date

2023

## Document Type

Dissertation

## Degree Name

Doctor of Philosophy

## Department

Mathematical Sciences

## Committee Chair

Maria Botelho

## Committee Member

David Grynkiewicz

## Committee Member

Anna Kamińska

## Committee Member

Máté Wierdl

## Committee Member

Pei-Kee Lin

## Abstract

In this thesis, we investigate certain ``preserver operators" from one Banach space into another. ``Preserver operators" are operators that leave unchanged one or more important properties of the domain space of the operator. Additionally, we explore operators related to or built from these preserver operators. Of importance will be the preserver operator known as an ``isometry," which are ``norm-persevering" operators, and the ``spectral projections" constructed from the elements contained in the spectrum of the isometry. The main results of this dissertation focus on projections (and more generally, $n$-potent operators) in the convex hull of a subgroup of the group of isometries on a Banach space, as well as projections (and $n$-potents) on the space of finite complex combinations of periodic or $n$-potent isometries (or more generally, for any arbitrary periodic or $n$-potent operator). Our progress extends research relating to generalized bi-circular projections, Hermitian operators, and surjective isometries with finite spectrum. Of interest are projections in the convex hull of a cyclic group $D$ generated by a periodic isometry $T$ with period equal to a positive integer $n \geq 2$. Let $Q = \sum_{k=1}^n \lambda_k T^k$ where $\lambda_k \in (0,1)$ and $\sum_{k=1}^n \lambda_k = 1$. Then, $Q$ is a projection if and only if $\lambda_k = \frac{1}{n}$ for each $k$. If $D$ is the general isometry group on the Banach space $AC(K,X)$ where $K$ is a non-trivial compact subset of $\R$ and $X$ is a strictly convex Banach space, then if $Q = \sum_{k=1}^n \lambda_k T^k$ where $\lambda_k \in (0,1)$, $\sum_{k=1}^n \lambda_k = 1$, and $\{ \lambda_1,...,\lambda_k \}$ is $2$-indecomposable, is a projection, then $P=I$ or $0$ (where $I$ is the identity operator on $X$).

## Library Comment

Dissertation or thesis originally submitted to ProQuest.

## Notes

Open Access

## Recommended Citation

Easley, Zachary, "Presever, $n$-potent, and Periodic Operators on Banach Spaces and Function Spaces" (2023). *Electronic Theses and Dissertations*. 3345.

https://digitalcommons.memphis.edu/etd/3345

## Comments

Data is provided by the student