Minimal invariant subspaces for composition operators

AbstractA remarkable result by Nordgren, Rosenthal and Wintrobe states that a positive answer to the Invariant Subspace Problem is equivalent to the statement that any minimal invariant subspace for a composition operator Cφ induced by a hyperbolic automorphism φ of the unit disc D in the Hardy space H2 is one dimensional. Motivated by this result, for f∈H2 we consider the space Kf, which is the closed subspace generated by the orbit of f. We obtain two results, one for functions with radial limit zero, and one for functions without radial limit zero, but tending to zero on a sequence of iterates. More precisely, for those functions f∈H2 with radial limit zero and continuous at the fixed points of φ, we provide a construction of a function g∈Kf such that f is a cluster point of the sequence of iterates {g∘φ−n}. In case f is in the disc algebra, we have Kg⊆Kf⊆span¯{g∘φn:n∈Z}. For a function f∈H2 tending to zero on a sequence of iterates {φn(z0)} at a point z0 with |z0|