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 fo 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 {zn } containing 0 and a subsequence {znj} satisfying the conditions g(znj)1, and
= 1.

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