On a projection from one co-invariant subspace onto another in character-automorphic Hardy space on a multiply connected domain

In a case of a theory in a unit disk the solution of a problem on the inver tibility of an orthogonal projection from one co-invariant subspace of the shift operator onto another turned out to be essential for the solution of the problem on the Riesz basis property of the reproducing kernels and in p articular for the solution of the problem on the basis of exponentials in L -2 space on a segment. In the present paper we are dealing with the similar problems in harmonic analysis on a finitely connected domain. Namely we ob tain necessary and sufficient conditions for the invertibility of an orthog onal projection from one co-invariant subspace of character-automorphic Har dy space in the domain onto another. The given condition has a form of a Mu ckenhoupt condition for a certain weight on the boundary of the domain, but essentially depends on a character. Namely, for two fixed character-automo rphic inner functions, which define the co-invariant subspaces, the project ion may be invertible for one character and not invertible for another.