### Abstract

Operators on function spaces acting by composition to the right with a fixed self-map ϕ are called composition operators. We denote them C_{ϕ}. Given ϕ, a hyperbolic disc automorphism, the composition operator C_{ϕ} on the Hilbert Hardy space H^{2} is considered. The bilateral cyclic invariant subspaces K_{f}, f ∈ H^{2}, of C_{ϕ} are studied, given their connection with the invariant subspace problem, which is still open for Hilbert space operators. We prove that nonconstant inner functions u induce non–minimal cyclic subspaces K_{u} if they have unimodular, orbital, cluster points. Other results about K_{u} when u is inner are obtained. If f ∈ H^{2} \ {0} has a bilateral orbit under C_{ϕ}, with Cesàro means satisfying certain boundedness conditions, we prove K_{f} is non–minimal invariant under C_{ϕ}. Other results proving the non–minimality of invariant subspaces of C_{ϕ} of type K_{f} when f is not an inner function are obtained as well.

Original language | English (US) |
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Title of host publication | Contemporary Mathematics |

Publisher | American Mathematical Society |

Pages | 247-262 |

Number of pages | 16 |

DOIs | |

Publication status | Published - Jan 1 2017 |

### Publication series

Name | Contemporary Mathematics |
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Volume | 699 |

ISSN (Print) | 0271-4132 |

ISSN (Electronic) | 1098-3627 |

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### Keywords

- Composition operators
- Invariant subspaces

### ASJC Scopus subject areas

- Mathematics(all)

### Cite this

*Contemporary Mathematics*(pp. 247-262). (Contemporary Mathematics; Vol. 699). American Mathematical Society. https://doi.org/10.1090/conm/699/14094