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Isometries of the Dirichlet Space among the Composition Operators
María J. Martín and Dragan Vukotić
Proceedings of the American Mathematical Society
Vol. 134, No. 6 (Jun., 2006), pp. 1701-1705
Published by: American Mathematical Society
Stable URL: http://www.jstor.org/stable/4098385
Page Count: 5
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We show that every composition operator which is an isometry of the Dirichlet space is induced by a univalent full map of the disk into itself that fixes the origin. This is an analogue of the Hardy space result for inner functions due to Nordgren. The proof relies on the Stone-Weierstrass theorem and the Riesz representation theorem.
Proceedings of the American Mathematical Society © 2006 American Mathematical Society