Date of Award

Summer 2019

Project Type

Dissertation

Department

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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