Applications of the Growth Characteristics Induced by the Spectral Distance

TL;DR

This paper explores the relationship between spectral distance and growth characteristics in Banach algebras, leading to a generalization of Gelfand's Power Boundedness Theorem and a characterization of equality of normal elements in $C^*$-algebras.

Contribution

It introduces a novel connection between spectral distance and entire map growth, generalizes Gelfand's theorem, and characterizes when two normal elements are equal in $C^*$-algebras.

Findings

Generalization of Gelfand's Power Boundedness Theorem

Normal elements are equal iff quasinilpotent equivalent in $C^*$-algebras

Spectral distance relates to growth characteristics of entire maps

Abstract

Let $A$ be a complex unital Banach algebra. Using a connection between the spectral distance and the growth characteristics of a certain entire map into $A$, we derive a generalization of Gelfand's famous Power Boundedness Theorem. Elaborating on these ideas, with the help of a Phragm\'{e}n-Lindel\"{o}f device for subharmonic functions, it is then shown, as the main result, that two normal elements of a $C^*$-algebra are equal if and only if they are quasinilpotent equivalent.