Date of Award

Summer 2019

Project Type

Dissertation

Program or Major

Mathematics

Degree Name

Doctor of Philosophy

First Advisor

Don Hadwin

Second Advisor

Mehmet Orhon

Third Advisor

Rita Hibschweiler

Abstract

\begin{Abstractpage}

\setlength{\baselineskip}{1.5\baselineskip} { Suppose $\mathcal{M}$ is a von Neumann algebra. An \textbf{operator range in

}$\mathcal{M}$ is the range of an operator in $\mathcal{M}$. When

$\mathcal{M}=B\left( H\right) $, the algebra of operators on a Hilbert space

$H$, R. Douglas and C. Foia\c{s} proved that if $S,T\in B\left( H\right) $,

and $T$ is not algebraic, and if $S$ leaves invariant every $T$-invariant

operator range, then $S=f\left( T\right) $ for some entire function $f$.

In the first part of this thesis, we prove versions of this result when $B\left( H\right) $ is replaced with a

factor von Neumann algebra $\mathcal{M}$ and $T$ is normal. Then using the direct integral theory, we extend our result to an arbitrary

von Neumann algebra.

In the second part of the thesis, we investigate the notion of \textbf{similarity dominance.}

Suppose $\mathcal{A}$ is a

unital Banach algebra and $S,T\in \mathcal{A}$. We say that $T$ sim-dominates

$S$ provided, for every $R>0$,%

\[

\sup \left( \left \{ \left \Vert A^{-1}SA\right \Vert :A\in \mathcal{A},\text{

}A\text{ invertible, }\left \Vert A^{-1}TA\right \Vert \leq R\right \} \right)

<\infty \text{.}%

\]

When $\mathcal{A}$ is the algebra $B\left( H\right) $, J. B. Conway and D.

Hadwin proved that $T$ sim-dominates $S$ implies $S=\varphi \left( T\right) $

for some entire function $\varphi$. We prove this for a large class of

operators in a type III factor von Neumann algebra.

We also prove, for any

unital Banach algebra $\mathcal{A}$, if $T$ sim-dominates $S$, then $S$ is in

the approximate double commutant of $T$ in $\mathcal{A}$.

Moreover, we prove that sim-domination is preserved under approximate similarity.

}

\end{Abstractpage}

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