Classification of Toeplitz Operators on Hardy Spaces of Bounded Domains in the Plane

We construct an orthonormal basis for the class of square integrable functions on bounded domains in the plane in terms of the classical kernel functions in potential theory. Then we generalize the results of Brown and Halmos about algebraic properties of Toeplitz operators and Laurent operators on the unit disc to general bounded domains. This is a complete classification of Laurent operators and Toeplitz operators for bounded domains.