On the minimal invariant subspaces of the hyperbolic composition operator

The composition operator induced by a hyperbolic Möbius transform ϕ on the classical Hardy space H² is considered. It is known that the invariant subspace problem for Hilbert space operators is equivalent to the fact that all the minimal invariant subspaces of this operator are one- dimensional. In connection with that we try to decide by the properties of a given function u in H² if the corresponding cyclic subspace is minimal or not. The main result is the following. If the radial limit of u is continuously extendable at one of the fixed points of ϕ and its value at the point is nonzero, then the cyclic subspace generated by u is minimal if and only if u is constant.