Angle between subspaces of analytic and antianalytic functions in weighted L2 space on the boundary of a multiply connected domain

The problem of describing those positive weights on the boundary Gamma of a finitely connected domain Omega for which the angle in the corresponding weighted L-2 space on Gamma between the linear space R(Omega) of all rational functions on C ($) over bar with poles outside of Clos Omega and the linear space R(Omega)_ = {f ($) over bar \f epsilon R(Omega)} of antianalytic rational functions, is a natural analog of the problem solved in the famous Helson-Szego theorem. In this paper we give a complete description tin terms of necessary and sufficient conditions) of those positive weights w on Gamma for which the angle in L-2(Gamma, w) between the spaces of analytic and antianalytic character-automorphic (non only single-valued) functions is nonzero. The given description is similar to the one in the Helson-Sarason Theorem for the unit disk: the "modified" weight obtained by factoring out finitely many zeros on Gamma must satisfy the Muckenhoupt condition, provided an additional condition on the divisor of these zeros and the character of the space under consideration holds.