### Abstract

We characterise the compact composition operators from any Mobius invariant Banach space to VMOA, the space of holomorphic functions on the unit disk U that have vanishing mean oscillation. We use this to obtain a characterisation of the compact composition operators from the Bloch space to VMOA. Finally, we study some properties of hyperbolic VMOA functions. We show that a function is hyperbolic VMOA if and only if it is the symbol of a compact composition operator from the Bloch space to VMOA.

Original language | English |
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Pages (from-to) | 1-19 |

Number of pages | 19 |

Journal | Bulletin of the Australian Mathematical Society |

Volume | 62 |

Issue number | 1 |

Publication status | Published - Aug 2000 |

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### ASJC Scopus subject areas

- Mathematics(all)

### Cite this

*Bulletin of the Australian Mathematical Society*,

*62*(1), 1-19.

**Composition operators on some Möbius invariant Banach spaces.** / Makhmutov, Shamil; Tjani, Maria.

Research output: Contribution to journal › Article

*Bulletin of the Australian Mathematical Society*, vol. 62, no. 1, pp. 1-19.

}

TY - JOUR

T1 - Composition operators on some Möbius invariant Banach spaces

AU - Makhmutov, Shamil

AU - Tjani, Maria

PY - 2000/8

Y1 - 2000/8

N2 - We characterise the compact composition operators from any Mobius invariant Banach space to VMOA, the space of holomorphic functions on the unit disk U that have vanishing mean oscillation. We use this to obtain a characterisation of the compact composition operators from the Bloch space to VMOA. Finally, we study some properties of hyperbolic VMOA functions. We show that a function is hyperbolic VMOA if and only if it is the symbol of a compact composition operator from the Bloch space to VMOA.

AB - We characterise the compact composition operators from any Mobius invariant Banach space to VMOA, the space of holomorphic functions on the unit disk U that have vanishing mean oscillation. We use this to obtain a characterisation of the compact composition operators from the Bloch space to VMOA. Finally, we study some properties of hyperbolic VMOA functions. We show that a function is hyperbolic VMOA if and only if it is the symbol of a compact composition operator from the Bloch space to VMOA.

UR - http://www.scopus.com/inward/record.url?scp=0040212692&partnerID=8YFLogxK

UR - http://www.scopus.com/inward/citedby.url?scp=0040212692&partnerID=8YFLogxK

M3 - Article

VL - 62

SP - 1

EP - 19

JO - Bulletin of the Australian Mathematical Society

JF - Bulletin of the Australian Mathematical Society

SN - 0004-9727

IS - 1

ER -