On the minimal invariant subspaces of the hyperbolic composition operator

Abstract:

The composition operator induced by a hyperbolic Möbius transform $\phi$ on the classical Hardy space ${H^2}$ 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 ${H^2}$ 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 $\phi$ and its value at the point is nonzero, then the cyclic subspace generated by $u$ is minimal if and only if $u$ is constant.