  Formally symmetric differential operators on weighted Hardy-Hilbert spaces
are analyzed, along with adjoint pairs of differential operators. Eigenvalue
problems for such operators are rather special, but include many of the
classical Riemann and Heun equations. Symmetric minimal operators are
characterized. A regular class whose leading coefficients have no zeros on the
unit circle are shown to be essentially self-adjoint. Eigenvalue asymptotics
are established. Some extensions to non-self-adjoint operators are also
considered.
