On the angles between subspaces, Muckenhoupt condition, and projection from one co-invariant subspace onto another in the theory of character-automorphic Hardy spaces on a multiply connected domain

It is known that in the case of the unit disk the invertibility of the orthogonal projection of one subspace of $H_2$ which is co-invariant with respect to the shift operator onto another such subspace is connected with the Helson–Szegő theorem and the Muckenhoupt condition. In the present paper, we consider the same problem in character-automorphic Hardy spaces on a finitely connected planar domain. The problem is reduced to estimating the angles between certain subspaces of the weighted $L_2$-space on the boundary of the domain. The answer is given in terms of the Muckenhoupt condition for certain weights.