Invariant subspace of composition operators on Hardy space

Abstract

We consider the invariant subspace of composition operators on Hardy space H p where the composition operators corresponding to a function ϕ that is a holomorphic self-map of D. Firstly, we discuss composition operators Cϕ on subspace H p α,β of Hardy space H p . We will explore the invariant subspaces for Cϕ in various special cases. Secondly, we consider Beurling type invariant subspace for Cϕ. When θ is a inner function, we prove that θH p is invariant for Cϕ if and only if θ • ϕ θ belongs to S(D). Thirdly, we obtain that z n H p is nontrivial invariant subspace for Deddends algebras D Cϕ when Cϕ is a compact composition operator and ϕ satisfies that ϕ(0) = 0 and ϕ ∞< 1.