THE HARDY-WEYL ALGEBRA

We study the algebra A generated by the Hardy operator H and the operator Mx of multiplication by x on L-2[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 Lambda and analytic on the interior of Lambda for a planar set Lambda = [-1,0]boolean OR D(1,1)<overline>, which we call the lollipop. We find a Toeplitz-like short exact sequence for the C & lowast;-algebra generated by A. We study the operator Z=H-M-x, show that its point spectrum is (-1,0]boolean OR D(1,1), and that the eigenvalues grow in multiplicity as the points move to 0 from the left.