### Abstract

The composition operator induced by a hyperbolic Möbius transform ϕ on the classical Hardy space H^{2} is considered. It is known that the invariant subspace problem for Hilbert space operators is equivalent to the fact that all the minimal invariant subspaces of this operator are one- dimensional. In connection with that we try to decide by the properties of a given function u in H^{2} if the corresponding cyclic subspace is minimal or not. The main result is the following. If the radial limit of u is continuously extendable at one of the fixed points of ϕ and its value at the point is nonzero, then the cyclic subspace generated by u is minimal if and only if u is constant.

Original language | English (US) |
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Pages (from-to) | 837-841 |

Number of pages | 5 |

Journal | Proceedings of the American Mathematical Society |

Volume | 119 |

Issue number | 3 |

DOIs | |

State | Published - 1993 |

Externally published | Yes |

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### ASJC Scopus subject areas

- Mathematics(all)
- Applied Mathematics

### Cite this

**On the minimal invariant subspaces of the hyperbolic composition operator.** / Matache, Valentin.

Research output: Contribution to journal › Article

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TY - JOUR

T1 - On the minimal invariant subspaces of the hyperbolic composition operator

AU - Matache, Valentin

PY - 1993

Y1 - 1993

N2 - The composition operator induced by a hyperbolic Möbius transform ϕ on the classical Hardy space H2 is considered. It is known that the invariant subspace problem for Hilbert space operators is equivalent to the fact that all the minimal invariant subspaces of this operator are one- dimensional. In connection with that we try to decide by the properties of a given function u in H2 if the corresponding cyclic subspace is minimal or not. The main result is the following. If the radial limit of u is continuously extendable at one of the fixed points of ϕ and its value at the point is nonzero, then the cyclic subspace generated by u is minimal if and only if u is constant.

AB - The composition operator induced by a hyperbolic Möbius transform ϕ on the classical Hardy space H2 is considered. It is known that the invariant subspace problem for Hilbert space operators is equivalent to the fact that all the minimal invariant subspaces of this operator are one- dimensional. In connection with that we try to decide by the properties of a given function u in H2 if the corresponding cyclic subspace is minimal or not. The main result is the following. If the radial limit of u is continuously extendable at one of the fixed points of ϕ and its value at the point is nonzero, then the cyclic subspace generated by u is minimal if and only if u is constant.

UR - http://www.scopus.com/inward/record.url?scp=84966212692&partnerID=8YFLogxK

UR - http://www.scopus.com/inward/citedby.url?scp=84966212692&partnerID=8YFLogxK

U2 - 10.1090/S0002-9939-1993-1152988-8

DO - 10.1090/S0002-9939-1993-1152988-8

M3 - Article

VL - 119

SP - 837

EP - 841

JO - Proceedings of the American Mathematical Society

JF - Proceedings of the American Mathematical Society

SN - 0002-9939

IS - 3

ER -