The Hardy-Weyl algebra

TL;DR

This paper investigates the structure of the Hardy-Weyl algebra generated by the Hardy operator and multiplication operator, revealing its quotient structure, spectral properties of a related operator, and establishing connections to function algebras on a specific planar set.

Contribution

It characterizes the quotient of the Hardy-Weyl algebra by compact operators, describes its relation to a function algebra on the lollipop set, and analyzes the spectral properties of the operator Z.

Findings

The quotient by compact operators is isomorphic to a function algebra on the lollipop set.

The operator Z has a point spectrum covering a specific planar set with eigenvalues of increasing multiplicity.

A Toeplitz-like short exact sequence is established for the generated C*-algebra.

Abstract

We study the algebra $\mathcal{A}$ generated by the Hardy operator $H$ and the operator $M_x$ of multiplication by $x$ on $L^2[0,1]$. We call $\mathcal{A}$ the Hardy-Weyl algebra. We show that its quotient by the compact operators is isomorphic to the algebra of functions that are continuous on $\Lambda$ and analytic on the interior of $\Lambda$ for a planar set $\Lambda$ = $[-1,0] \cup \bar{ \mathbb{D}(1,1)}$, which we call the lollipop. We find a Toeplitz-like short exact sequence for the $C^*$-algebra generated by $\mathcal{A}$. We study the operator $Z = H - M_x$, show that its point spectrum is $(-1,0] \cup \mathbb{D}(1,1)$, and that the eigenvalues grow in multiplicity as the points move to $0$ from the left.