Isometries of the Dirichlet space among the composition operators
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- by María J. Martín and Dragan Vukotić PDF
- Proc. Amer. Math. Soc. 134 (2006), 1701-1705 Request permission
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
- 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 786732, DOI 10.1215/ijm/1256045634
- J. Arazy, S. D. Fisher, and J. Peetre, Möbius invariant function spaces, J. Reine Angew. Math. 363 (1985), 110–145. MR 814017, DOI 10.1007/BFb0078341
- J. A. Cima and W. R. Wogen, On isometries of the Bloch space, Illinois J. Math. 24 (1980), no. 2, 313–316. MR 575069
- Carl C. Cowen and Barbara D. MacCluer, Composition operators on spaces of analytic functions, Studies in Advanced Mathematics, CRC Press, Boca Raton, FL, 1995. MR 1397026
- Peter Duren and Alexander Schuster, Bergman spaces, Mathematical Surveys and Monographs, vol. 100, American Mathematical Society, Providence, RI, 2004. MR 2033762, DOI 10.1090/surv/100
- Eva A. Gallardo-Gutiérrez and Alfonso Montes-Rodríguez, Adjoints of linear fractional composition operators on the Dirichlet space, Math. Ann. 327 (2003), no. 1, 117–134. MR 2005124, DOI 10.1007/s00208-003-0442-9
- William Hornor and James E. Jamison, Isometries of some Banach spaces of analytic functions, Integral Equations Operator Theory 41 (2001), no. 4, 410–425. MR 1857800, DOI 10.1007/BF01202102
- María J. Martín and Dragan Vukotić, Norms and spectral radii of composition operators acting on the Dirichlet space, J. Math. Anal. Appl. 304 (2005), no. 1, 22–32. MR 2124646, DOI 10.1016/j.jmaa.2004.09.005
- Eric A. Nordgren, Composition operators, Canadian J. Math. 20 (1968), 442–449. MR 223914, DOI 10.4153/CJM-1968-040-4
- Joel H. Shapiro, Composition operators and classical function theory, Universitext: Tracts in Mathematics, Springer-Verlag, New York, 1993. MR 1237406, DOI 10.1007/978-1-4612-0887-7
- Joel H. Shapiro, What do composition operators know about inner functions?, Monatsh. Math. 130 (2000), no. 1, 57–70. MR 1762064, DOI 10.1007/s006050050087
- C. Voas, Toeplitz Operators and Univalent Functions, Ph.D. Thesis, University of Virginia, 1980.
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
- Email: mjmartin@eco.uc3m.es, mjose.martin@uam.es
- Dragan Vukotić
- Affiliation: Departamento de Matemáticas, Universidad Autónoma de Madrid, 28049 Madrid, Spain
- Email: dragan.vukotic@uam.es
- 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
- © Copyright 2005
American Mathematical Society
The copyright for this article reverts to public domain 28 years after publication. - Journal: Proc. Amer. Math. Soc. 134 (2006), 1701-1705
- MSC (2000): Primary 47B33, 31C25
- DOI: https://doi.org/10.1090/S0002-9939-05-08182-7
- MathSciNet review: 2204282