  By working with all collection of all the Sarason Hilbert Hardy spaces for
the annulus algebra an improvement to the results of Aryana and Clancey on
eigenvalues of self adjoint Toeplitz operators on an annulus is obtained. The
ideas are applied to Toeplitz operators on the Neil algebra. These examples may
provide a template for a general theory of Toeplitz operators with respect to
an algebra.
