THE HARDY–WEYL ALGEBRA

We study the algebra A generated by the Hardy operator H and the operator M_x of multiplication by x on L²[0, 1]. We call 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 Λ and analytic on the interior of Λ for a planar set Λ = [−1, 0] ∪ D(1, 1), which we call the lollipop. We find a Toeplitz-like short exact sequence for the C^∗-algebra generated by A. We study the operator Z = H − M_x, show that its point spectrum is (−1, 0] ∪D(1, 1), and that the eigenvalues grow in multiplicity as the points move to 0 from the left.