*Bull. Austral. Math. Soc.* 72(2) pp.283--290, 2005.

# Characterisation of the isometric composition operators on the Bloch space

## Flavia Colonna |

I wish to dedicate this article to Professor Maurice
Heins for his ninetieth birthday.

I owe him a debt of gratitude for his great lectures which deeply
stimulated my passion for complex analysis.

As a thesis advisor, he was always very patient and generous with
his time.

## Abstract

In this paper, we characterise the analytic
functions
mapping the open unit disk into itself whose induced composition operator

*C*_{}:*f**f*`o`is an isometry on the Bloch space. We show that such functions are either rotations of the identity function or have a factorisation =*gB*where*g*is a non-vanishing analytic function from into the closure of , and*B*is an infinite Blaschke product whose zeros form a sequence {*z*_{n}} containing 0 and a subsequence {*z*_{n}j} satisfying the conditions*g*(*z*_{n}j)1, and
= 1.

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(Metadata: XML, RSS, BibTeX) | MathSciNet: MR2183409 | Z'blatt-MATH: 02246390 |

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ISSN 0004-9727