Singly Generated Radical Operator Algebras

TL;DR

This paper investigates two nonselfadjoint operator algebras, analyzing their ideal structures and norm properties using Fourier analysis and alternative methods, highlighting differences due to the presence or absence of gauge actions.

Contribution

It introduces new techniques for studying the ideal structures of weighted shift and Volterra operator algebras, especially addressing cases without gauge actions.

Findings

Weighted shift algebra admits Fourier analysis for ideal study.

Volterra operator algebra requires alternative methods due to lack of gauge action.

Both algebras are the norm closures of polynomials in their operators.

Abstract

We examine two nonselfadjoint operator algebras: the weighted shift algebra, and the Volterra operator algebra. In both cases, the operator algebra is the norm closure of the polynomials in the operator norm. In the case of the weighted shift algebra, the existence of a gauge action allows us to apply Fourier analysis to study the ideals of the algebra. In the case of the Volterra operator algebra, there is no gauge action, and other methods are needed to study the norm structure and the ideals.