@article {Colonna2005,
 author="Flavia Colonna",
 title={Characterisation of the isometric composition operators on the Bloch space},
 journal="Bull. Austral. Math. Soc.",
 fjournal={Bulletin of the Australian Mathematical Society},
 volume="72",
 year="2005",
 number="2",
 pages="283--290",
 issn="0004-9727",
 coden="ALNBAB",
 language="English",
 date="3rd May, 2005",
 classmath="30D45, 47B33, 47A30",
 publisher={AMPAI, Australian Mathematical Society},
 MRnumber="MR2183409",
 ZBLnumber="02246390",
 url="http://www.austms.org.au/Publ/Bulletin/V72P2/722-5133-Colonna/index.shtml",
 acknowledgement={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 $\f $ mapping the open unit disk $\D $ into itself whose induced composition operator $C_\f : f\mapsto f\circ \f $ is an isometry on the Bloch space. We show that such functions are either rotations of the identity function or have a factorisation $\f =gB$ where $g$ is a non-vanishing analytic function from $\D $ into the closure of $\D $, 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 $\bigl |g(z_{n_{j}})\bigr |\to 1$, and $$\lim _{j\to \infty }\prod _{k\ne n_j}\Bigl |\frac {z_{n_j}-z_k}{1-\overline {z_{n_j}}z_k}\Bigr |=1.$$ }
}