# Journal of Operator Theory

Volume 91, Issue 2, Spring 2024 pp. 521-544.

The Hardy-Weyl algebra

**Authors**:
Jim Agler (1), John E. McCarthy (2)

**Author institution:** (1) Department of Mathematics, University of California at San Diego, 2985
Muir Ln, La Jolla, CA 92093, U.S.A.

(2) Department of Mathematics, Washington University in St. Louis, 1
Brookings Drive, St. Louis MO 63130, U.S.A.

**Summary: ** 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 $\Lambda$ = $[-1,0] \cup \overline{ \mathbb{D}(1,1)}$, 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.

**DOI: **http://dx.doi.org/10.7900/jot.2022jun15.2387

**Keywords: **Hardy operator, Hardy-Weyl algebra, lollipop algebra

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