### 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 | |

State | Published - Jan 1 2015 |

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### Keywords

- Composition operator
- Invariant subspaces

### ASJC Scopus subject areas

- Algebra and Number Theory

### Cite this

**Invariant subspaces of composition operators.** / Matache, Valentin.

Research output: Contribution to journal › Article

*Journal of Operator Theory*, vol. 73, no. 1, pp. 243-264. https://doi.org/10.7900/jot.2013nov14.2041

}

TY - JOUR

T1 - Invariant subspaces of composition operators

AU - Matache, Valentin

PY - 2015/1/1

Y1 - 2015/1/1

N2 - The invariant subspace lattices of composition operators acting on H2, 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 Mz (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 Cϕ, the composition operator induced by ϕ, maps certain subspaces in the invariant subspace lattice of Mz into other such spaces. A similar characterization of the existence of angular derivatives of ϕ, this time in terms of Aϕ, the Aleksandrov operator induced by ϕ, is obtained.

AB - The invariant subspace lattices of composition operators acting on H2, 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 Mz (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 Cϕ, the composition operator induced by ϕ, maps certain subspaces in the invariant subspace lattice of Mz into other such spaces. A similar characterization of the existence of angular derivatives of ϕ, this time in terms of Aϕ, the Aleksandrov operator induced by ϕ, is obtained.

KW - Composition operator

KW - Invariant subspaces

UR - http://www.scopus.com/inward/record.url?scp=84925273977&partnerID=8YFLogxK

UR - http://www.scopus.com/inward/citedby.url?scp=84925273977&partnerID=8YFLogxK

U2 - 10.7900/jot.2013nov14.2041

DO - 10.7900/jot.2013nov14.2041

M3 - Article

AN - SCOPUS:84925273977

VL - 73

SP - 243

EP - 264

JO - Journal of Operator Theory

JF - Journal of Operator Theory

SN - 0379-4024

IS - 1

ER -