Invariant subspaces of parabolic self-maps in the Hardy space

We provide a precise description of the lattice of invariant subspaces of composition operators acting on the classical Hardy space, whose inducing symbol is a parabolic non-automorphism. This is achieved with an explicit isomorphism between the Hardy space and the Sobolev Banach algebra $W^{1,2}[0,\infty)$ that induces a bijection between the lattice of the composition operator and the closed ideals of $W^{1,2}[0,\infty)$. In particular, each invariant subspace of parabolic non-automorphism composition operator always consists of the closed span of a set of eigenfunctions. As a consequence, such composition operators have no non-trivial reducing subspaces. For the sake of completeness, we also include a characterization of the closed ideals of the Banach algebra $W^{1,2}[0,\infty)$. Although such a characterization is known, the proof we provide here is somehow different.

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Published 1 January 2010