Invertible composition operators on \(H^ p\) (Q1101934)

The author considers the algebras of composition operators in the form \((C_{\Phi}f)(z) = f(\Phi(z))\), \(| z| <1\) \((C_{\Phi}:H^p(D)\to H^p(D))\), where \(\Phi\) is an analytic function that maps the unit disk \(D\) into itself. The main result of this paper concerns the structure of parabolic and hyperbolic composition operators and proves that the strongly closed algebra generated by a single invertible composition operator on \(H^ 2(D)\) is reflexive. Theorem 1 shows that every hyperbolic composition operator on \(H^ 2\) is cosubnormal and has universal translates. It follows that every operator on a Hilbert space has an invariant subspace if and only if the minimal invariant subspaces of the operator \(C_{\Phi}\) for \(\Phi(z)=(2z-1)/2-z\) are one dimensional.