Mathematics Ph.D. Dissertations

What Certain Norms Say About Spectra

Date of Award

2022

Document Type

Dissertation

Degree Name

Doctor of Philosophy (Ph.D.)

Department

Mathematics

First Advisor

Alexander Izzo (Committee Chair)

Second Advisor

Alicia Mrachko (Other)

Third Advisor

Kit Chan (Committee Member)

Fourth Advisor

Mihai Staic (Committee Member)

Abstract

The spectral radius formula is a well known way that the norms of powers of a Banach algebra element give information about its spectrum, giving the radius of the smallest closed disc in the complex plane centered at the origin containing the spectrum. This dissertation explores what beyond the spectral radius formula the norms of powers say about the spectrum. We also consider the information about the joint spectrum of several elements in a commutative Banach algebra contained in the norms of products of powers of the elements. A condition on the norms of powers of a single Banach algebra element for the spectrum to necessarily contain the origin is presented. As a generalization of this result, we find the maximum possible inner radius for the smallest annulus centered at the origin containing the spectrum of a Banach algebra element, given the norms of powers of the element. The results on the joint spectrum of several commutative Banach algebra elements begin with a generalization of the spectral radius formula, giving the smallest polynomially convex circled set containing the joint spectrum. Additionally, we present a several variable analogue to the one variable result on annuli, in that we find the largest rationally convex connected circled set that must be contained in the smallest rationally convex connected circled set containing the joint spectrum of several elements in a commutative Banach algebra, given the norms of products of powers of the elements.

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