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Composition operators with maximal norm on weighted Bergman spaces


Authors: Brent J. Carswell and Christopher Hammond
Journal: Proc. Amer. Math. Soc. 134 (2006), 2599-2605
MSC (2000): Primary 47B33
DOI: https://doi.org/10.1090/S0002-9939-06-08271-2
Published electronically: February 17, 2006
MathSciNet review: 2213738
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Abstract: We prove that any composition operator with maximal norm on one of the weighted Bergman spaces $ A^{2}_{\alpha}$ (in particular, on the space $ A^{2}=A^{2}_{0}$) is induced by a disk automorphism or a map that fixes the origin. This result demonstrates a major difference between the weighted Bergman spaces and the Hardy space $ H^{2}$, where every inner function induces a composition operator with maximal norm.


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  • 1. P. S. Bourdon, E. E. Fry, C. Hammond, and C. H. Spofford, Norms of linear-fractional composition operators, Trans. Amer. Math. Soc. 356 (2004), 2459-2480. MR 2048525 (2004m:47045)
  • 2. C. C. Cowen, Linear fractional composition operators on $ H^{2}$, Integral Equations Operator Theory 11 (1988), 151-160. MR 0928479 (89b:47044)
  • 3. C. C. Cowen and T. L. Kriete, Subnormality and composition operators on $ H^2$, J. Funct. Anal. 81 (1988), 298-319. MR 0971882 (90c:47055)
  • 4. C. C. Cowen and B. D. MacCluer, Composition Operators on Spaces of Analytic Functions, CRC Press, Boca Raton, 1995. MR 1397026 (97i:47056)
  • 5. P. L. Duren, Theory of $ H^{p}$ Spaces, Academic Press, New York, 1970. MR 0268655 (42:3552)
  • 6. P. Duren and A. Schuster, Bergman Spaces, American Mathematical Society, Providence, 2004. MR 2033762 (2005c:30053)
  • 7. C. Hammond, On the norm of a composition operator with linear fractional symbol, Acta Sci. Math. (Szeged) 69 (2003), 813-829. MR 2034210 (2004m:47049)
  • 8. C. Hammond, On the norm of a composition operator, Ph.D. thesis, University of Virginia, 2003.
  • 9. H. Hedenmalm, B. Korenblum, and K. Zhu, Theory of Bergman Spaces, Springer-Verlag, New York, 2000. MR 1758653 (2001c:46043)
  • 10. P. R. Hurst, Relating composition operators on different weighted Hardy spaces, Arch. Math. (Basel) 68 (1997), 503-513. MR 1444662 (98c:47040)
  • 11. M. J. Martín and D. Vukotic, Isometries of the Dirichlet space among the composition operators, Proc. Amer. Math. Soc., to appear.
  • 12. V. Matache, A short proof of a characterization of inner functions in terms of the composition operators they induce, Rocky Mountain J. Math. 35 (2005), 1723-1726.
  • 13. E. A. Nordgren, Composition operators, Canad. J. Math. 20 (1968), 442-449. MR 0223914 (36:6961)
  • 14. P. Poggi-Corradini, The essential norm of composition operators revisited, Studies on Composition Operators (Laramie, 1996), 167-173, Contemp. Math., 213, Amer. Math. Soc., Providence, 1998. MR 1601104 (98m:47046)
  • 15. D. B. Pokorny and J. E. Shapiro, Continuity of the norm of a composition operator, Integral Equations Operator Theory 45 (2003), 351-358. MR 1965901 (2004b:47047)
  • 16. A. E. Richman, Subnormality and composition operators on the Bergman space, Integral Equations Operator Theory 45 (2003), 105-124. MR 1952344 (2004c:47050)
  • 17. A. E. Richman, Composition operators with complex symbol having subnormal adjoint, Houston J. Math. 29 (2003), 371-384. MR 1987582 (2004i:47049)
  • 18. J. H. Shapiro, The essential norm of a composition operator, Annals Math. 125 (1987), 375-404. MR 0881273 (88c:47058)
  • 19. J. H. Shapiro, What do composition operators know about inner functions?, Monatsh. Math. 130 (2000), 57-70. MR 1762064 (2001a:47029)
  • 20. S. M. Shimorin, Factorization of analytic functions in weighted Bergman spaces (Russian), Algebra i Analiz 5 (1993), 155-177; translation in St. Petersburg Math. J. 5 (1994), 1005-1022. MR 1263318 (95j:30032)
  • 21. S. M. Shimorin, On a family of conformally invariant operators (Russian), Algebra i Analiz 7 (1995), 133-158; translation in St. Petersburg Math. J. 7 (1996), 287-306. MR 1347516 (96i:47010)
  • 22. D. Vukotic, On norms of composition operators acting on Bergman spaces, J. Math. Anal. Appl. 291 (2004), 189-202. MR 2034066 (2004m:30058)
  • 23. D. Vukotic, Corrigendum to ``On norms of composition operators acting on Bergman spaces" [J. Math. Anal. Appl. 291 (2004), 189-202. MR 2034066 (2004m:30058)], J. Math. Anal. Appl. 311 (2005), 377-380. MR 2165484

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Additional Information

Brent J. Carswell
Affiliation: Department of Mathematics, Allegheny College, Meadville, Pennsylvania 16335
Email: brent.carswell@allegheny.edu

Christopher Hammond
Affiliation: Department of Mathematics and Computer Science, Connecticut College, New London, Connecticut 06320
Email: cnham@conncoll.edu

DOI: https://doi.org/10.1090/S0002-9939-06-08271-2
Keywords: Composition operator, norm, essential norm
Received by editor(s): February 2, 2005
Received by editor(s) in revised form: March 21, 2005
Published electronically: February 17, 2006
Communicated by: Joseph A. Ball
Article copyright: © Copyright 2006 American Mathematical Society
The copyright for this article reverts to public domain 28 years after publication.

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