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Isometries of the Dirichlet space among the composition operators

Authors: María J. Martín and Dragan Vukotic
Journal: Proc. Amer. Math. Soc. 134 (2006), 1701-1705
MSC (2000): Primary 47B33, 31C25
Published electronically: December 2, 2005
MathSciNet review: 2204282
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Abstract: 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.

References [Enhancements On Off] (What's this?)

  • 1. J. Arazy and S.D. Fisher, The uniqueness of the Dirichlet space among Möbius-invariant Hilbert spaces, Illinois J. Math. 29 (1985), no. 3, 449-462. MR 0786732 (86j:30072)
  • 2. J. Arazy, S.D. Fisher, and J. Peetre, Möbius invariant function spaces, J. Reine Angew. Math. 363 (1985), 110-145. MR 0814017 (87f:30104)
  • 3. J.A. Cima and W.R. Wogen, On isometries of the Bloch space, Illinois J. Math. 24 (1980), no. 2, 313-316. MR 0575069 (82m:30052)
  • 4. C. Cowen and B. MacCluer, Composition Operators on Spaces of Analytic Functions, Studies in Advanced Mathematics, CRC Press, Boca Raton, 1995. MR 1397026 (97i:47056)
  • 5. P.L. Duren and A.P. Schuster, Bergman Spaces, Mathematical Surveys and Monographs 100, American Mathematical Society, Providence, RI, 2004. MR 2033762 (2005c:30053)
  • 6. E.A. Gallardo-Gutiérrez and A. Montes-Rodríguez, Adjoints of linear fractional compostion operators on the Dirichlet space, Math. Ann., 327 (2003), no. 1, 117-134. MR 2005124 (2004h:47036)
  • 7. W. Hornor and J.E. Jamison, Isometries of some Banach spaces of analytic functions, Integral Equations Operator Theory 41 (2001), no. 4, 410-425. MR 1857800 (2002h:46035)
  • 8. M.J. Martín and D. Vukotic, Norms and spectral radii of composition operators acting on the Dirichlet space, J. Math. Anal. Appl. 304 (2005), 22-32. MR 2124646
  • 9. E.A. Nordgren, Composition operators, Canad. J. Math. 20 (1968), 442-449. MR 0223914 (36:6961)
  • 10. J.H. Shapiro, Composition Operators and Classical Function Theory, Springer-Verlag, New York, 1993. MR 1237406 (94k:47049)
  • 11. J.H. Shapiro, What do composition operators know about inner functions?, Monatsh. Math. 130 (2000), 57-70. MR 1762064 (2001a:47029)
  • 12. C. Voas, Toeplitz Operators and Univalent Functions, Ph.D. Thesis, University of Virginia, 1980.

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Additional Information

María J. Martín
Affiliation: Departamento de Economía, Universidad Carlos III de Madrid, Calle Madrid 126, 28903 Getafe (Madrid), Spain
Address at time of publication: Departamento de Matemáticas, Universidad Autónoma de Madrid, 28049 Madrid, Spain

Dragan Vukotic
Affiliation: Departamento de Matemáticas, Universidad Autónoma de Madrid, 28049 Madrid, Spain

Received by editor(s): December 14, 2004
Received by editor(s) in revised form: January 10, 2005
Published electronically: December 2, 2005
Additional Notes: Both authors were supported by MCyT grant BFM2003-07294-C02-01, Spain.
Communicated by: Joseph A. Ball
Article copyright: © Copyright 2005 American Mathematical Society
The copyright for this article reverts to public domain 28 years after publication.

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