### Abstract

The invariant subspace lattices of composition operators acting on H^{2}, the Hilbert-Hardy space over the unit disc, are characterized in select cases. The lattice of all spaces left invariant by both a composition operator and the unilateral shift M_{z} (the multiplication operator induced by the coordinate function), is shown to be nontrivial and is completely described in particular cases. Given an analytic selfmap ϕ of the unit disc, we prove that ϕ has an angular derivative at some point on the unit circle if and only if C_{ϕ}, the composition operator induced by ϕ, maps certain subspaces in the invariant subspace lattice of M_{z} into other such spaces. A similar characterization of the existence of angular derivatives of ϕ, this time in terms of A_{ϕ}, the Aleksandrov operator induced by ϕ, is obtained.

Original language | English (US) |
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Pages (from-to) | 243-264 |

Number of pages | 22 |

Journal | Journal of Operator Theory |

Volume | 73 |

Issue number | 1 |

DOIs | |

Publication status | Published - Jan 1 2015 |

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### Keywords

- Composition operator
- Invariant subspaces

### ASJC Scopus subject areas

- Algebra and Number Theory