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Dissertation/Thesis Abstract

Non-Euclidean Metric on the Resolvent Set
by Tran, Mai Thi Thuy, Ph.D., State University of New York at Albany, 2019, 61; 27664256
Abstract (Summary)

For a bounded linear operator $A$ on a complex Hilbert space $\mathcal{H}$, the Douglas-Yang metric on the resolvent set $\rho(A)$ is defined by the metric function $g_{\vec{x}}(z)=\left \| \big(A -z I\big)^{-1} \vec{x} \right \|^2$, where $\vec{x} \in \mathcal{H}$ with $\left \| \vec{x} \right \|=1$.

The Douglas-Yang metric provide the opportunity to see what particular properties relating to the operator $A$ can be determine by the use of geometry. We look at how the arc length, curvature, and geodesic equations are defined with respect to the Douglas-Yang metric in the case when $A$ is nilpotent operator and the unilateral shift operator. When $A$ is the unilateral shift operator on the Hardy space $H^2(\mathbb{D})$, the metric function $g_{f}(z)$ with respect to the Douglas-Yang metric is dependent on the chosen function $f \in H(\mathbb{D})$. We determine the extremal values of the arc length of the circle $C_{r}$ centered at the origin with radius $r$ with respect to the choice of $f$ in the Douglas-Yang metric. We showed that the maximal value is reached precisely when $f$ is an inner function.

Other considerations include defining the metric function $f_{A}(z)$ by using trace, extremal values of the Ricci curvature with respect to the Douglas-Yang metric. And it appears that its extremal value is related to the normalized trace of the matrix $A$.

Indexing (document details)
Advisor: Yang, Rongwei
Commitee: Stessin, Michael, Zhu, Kehe, Kwon, Hyun-Kyoung
School: State University of New York at Albany
Department: Mathematics and Statistics
School Location: United States -- New York
Source: DAI-B 81/8(E), Dissertation Abstracts International
Subjects: Mathematics
Keywords: Extremal equation, Functional analysis, Geometry, Nilpotent operator, Resolvent set, Unilateral shift operator
Publication Number: 27664256
ISBN: 9781658405119
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