Composition operators with maximal norm on weighted Bergman spaces

Carswell, Brent; Hammond, Christopher

doi:10.1090/S0002-9939-06-08271-2

Composition operators with maximal norm on weighted Bergman spaces
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by Brent J. Carswell and Christopher Hammond

Proc. Amer. Math. Soc. 134 (2006), 2599-2605

DOI: https://doi.org/10.1090/S0002-9939-06-08271-2

Published electronically: February 17, 2006

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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.

References

P. S. Bourdon, E. E. Fry, C. Hammond, and C. H. Spofford, Norms of linear-fractional composition operators, Trans. Amer. Math. Soc. 356 (2004), no. 6, 2459–2480. MR 2048525, DOI 10.1090/S0002-9947-03-03374-9
Carl C. Cowen, Linear fractional composition operators on $H^2$, Integral Equations Operator Theory 11 (1988), no. 2, 151–160. MR 928479, DOI 10.1007/BF01272115
Carl C. Cowen and Thomas L. Kriete III, Subnormality and composition operators on $H^2$, J. Funct. Anal. 81 (1988), no. 2, 298–319. MR 971882, DOI 10.1016/0022-1236(88)90102-4
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 L. Duren, Theory of $H^{p}$ spaces, Pure and Applied Mathematics, Vol. 38, Academic Press, New York-London, 1970. MR 0268655
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
Christopher Hammond, On the norm of a composition operator with linear fractional symbol, Acta Sci. Math. (Szeged) 69 (2003), no. 3-4, 813–829. MR 2034210
C. Hammond, On the norm of a composition operator, Ph.D. thesis, University of Virginia, 2003.
Haakan Hedenmalm, Boris Korenblum, and Kehe Zhu, Theory of Bergman spaces, Graduate Texts in Mathematics, vol. 199, Springer-Verlag, New York, 2000. MR 1758653, DOI 10.1007/978-1-4612-0497-8
Paul R. Hurst, Relating composition operators on different weighted Hardy spaces, Arch. Math. (Basel) 68 (1997), no. 6, 503–513. MR 1444662, DOI 10.1007/s000130050083
M. J. Martín and D. Vukotić, Isometries of the Dirichlet space among the composition operators, Proc. Amer. Math. Soc., to appear.
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.
Eric A. Nordgren, Composition operators, Canadian J. Math. 20 (1968), 442–449. MR 223914, DOI 10.4153/CJM-1968-040-4
Pietro Poggi-Corradini, The essential norm of composition operators revisited, Studies on composition operators (Laramie, WY, 1996) Contemp. Math., vol. 213, Amer. Math. Soc., Providence, RI, 1998, pp. 167–173. MR 1601104, DOI 10.1090/conm/213/02858
David B. Pokorny and Jonathan E. Shapiro, Continuity of the norm of a composition operator, Integral Equations Operator Theory 45 (2003), no. 3, 351–358. MR 1965901, DOI 10.1007/s000200300010
Alexander E. Richman, Subnormality and composition operators on the Bergman space, Integral Equations Operator Theory 45 (2003), no. 1, 105–124. MR 1952344, DOI 10.1007/BF02789595
Alexander E. Richman, Composition operators with complex symbol having subnormal adjoint, Houston J. Math. 29 (2003), no. 2, 371–384. MR 1987582
Joel H. Shapiro, The essential norm of a composition operator, Ann. of Math. (2) 125 (1987), no. 2, 375–404. MR 881273, DOI 10.2307/1971314
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
S. M. Shimorin, Factorization of analytic functions in weighted Bergman spaces, Algebra i Analiz 5 (1993), no. 5, 155–177 (Russian); English transl., St. Petersburg Math. J. 5 (1994), no. 5, 1005–1022. MR 1263318
S. M. Shimorin, On a family of conformally invariant operators, Algebra i Analiz 7 (1995), no. 2, 133–158 (Russian); English transl., St. Petersburg Math. J. 7 (1996), no. 2, 287–306. MR 1347516
Dragan Vukotić, On norms of composition operators acting on Bergman spaces, J. Math. Anal. Appl. 291 (2004), no. 1, 189–202. MR 2034066, DOI 10.1016/j.jmaa.2003.10.025
Dragan Vukotić, On norms of composition operators acting on Bergman spaces, J. Math. Anal. Appl. 291 (2004), no. 1, 189–202. MR 2034066, DOI 10.1016/j.jmaa.2003.10.025