Extremal Solutions of the Two-DimensionalL-Problem of Moments

By using the theory of the principal function of a hyponormal operator with rank-one self-commutator, one classifies some extremal points of the solutions of the truncatedL-problem of moments in two real variables. These extremal elements, called degenerated solutions of theL-problem, are proved to be in a natural bijective correspondence with the bounded quadrature domains in the complex plane. The main result of the paper characterizes these degenerated solutions from three distinct perspectives: complex analysis, moment problems, and operator theory. The interplay between the three points of view, each with specific techniques, constitutes the main body of the paper.