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

Characterisation of the isometric composition operators on the Bloch space

Flavia Colonna

Received: 3rd May, 2005

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.


In this paper, we characterise the analytic functions $ \varphi $ mapping the open unit disk $ \Delta $ into itself whose induced composition operator C$\scriptstyle \varphi $ : f $ \mapsto $ fo$ \varphi $ is an isometry on the Bloch space. We show that such functions are either rotations of the identity function or have a factorisation $ \varphi $ = gB where g is a non-vanishing analytic function from $ \Delta $ into the closure of $ \Delta $, and B is an infinite Blaschke product whose zeros form a sequence {zn } containing 0 and a subsequence {znj} satisfying the conditions $ \bigl \vert $g(znj)$ \bigr \vert $$ \to $1, and
$\displaystyle \lim _{{j\to \infty }}^{}$$\displaystyle \prod _{{k\ne n_j}}^{}$$\displaystyle \Bigl \vert $$\displaystyle {\frac {{z_{n_{j}}-z_k}}{{1-\overline {z_{n_{j}}}z_k}}}$ $\displaystyle \Bigr \vert $ = 1.

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


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