Reproducing kernels, de Branges-Rovnyak spaces, and norms of weighted composition operators
Author:
Michael T. Jury
Journal:
Proc. Amer. Math. Soc. 135 (2007), 3669-3675
MSC (2000):
Primary 47B33; Secondary 47B32, 46E22
DOI:
https://doi.org/10.1090/S0002-9939-07-08931-9
Published electronically:
August 15, 2007
MathSciNet review:
2336583
Full-text PDF Free Access
Abstract | References | Similar Articles | Additional Information
Abstract: We prove that the norm of a weighted composition operator on the Hardy space of the disk is controlled by the norm of the weight function in the de Branges-Rovnyak space associated to the symbol of the composition operator. As a corollary we obtain a new proof of the boundedness of composition operators on
and recover the standard upper bound for the norm. Similar arguments apply to weighted Bergman spaces. We also show that the positivity of a generalized de Branges-Rovnyak kernel is sufficient for the boundedness of a given composition operator on the standard function spaces on the unit ball.
- 1. Jim Agler and John E. McCarthy, Pick interpolation and Hilbert function spaces, Graduate Studies in Mathematics, vol. 44, American Mathematical Society, Providence, RI, 2002. MR 1882259
- 2. Joseph A. Cima, Charles S. Stanton, and Warren R. Wogen, On boundedness of composition operators on 𝐻²(𝐵₂), Proc. Amer. Math. Soc. 91 (1984), no. 2, 217–222. MR 740174, https://doi.org/10.1090/S0002-9939-1984-0740174-7
- 3. Manuel D. Contreras and Alfredo G. Hernández-Díaz, Weighted composition operators on Hardy spaces, J. Math. Anal. Appl. 263 (2001), no. 1, 224–233. MR 1864316, https://doi.org/10.1006/jmaa.2001.7610
- 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
- 5. Carl C. Cowen and Barbara D. MacCluer, Linear fractional maps of the ball and their composition operators, Acta Sci. Math. (Szeged) 66 (2000), no. 1-2, 351–376. MR 1768872
- 6. Louis de Branges and James Rovnyak, Square summable power series, Holt, Rinehart and Winston, New York-Toronto, Ont.-London, 1966. MR 0215065
- 7. Barbara D. MacCluer, Compact composition operators on 𝐻^{𝑝}(𝐵_{𝑁}), Michigan Math. J. 32 (1985), no. 2, 237–248. MR 783578, https://doi.org/10.1307/mmj/1029003191
- 8. Eric A. Nordgren, Composition operators, Canadian J. Math. 20 (1968), 442–449. MR 223914, https://doi.org/10.4153/CJM-1968-040-4
- 9. Donald Sarason, Sub-Hardy Hilbert spaces in the unit disk, University of Arkansas Lecture Notes in the Mathematical Sciences, vol. 10, John Wiley & Sons, Inc., New York, 1994. A Wiley-Interscience Publication. MR 1289670
- 10. Serguei Shimorin, Commutant lifting and factorization of reproducing kernels, J. Funct. Anal. 224 (2005), no. 1, 134–159. MR 2139107, https://doi.org/10.1016/j.jfa.2004.08.011
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Additional Information
Michael T. Jury
Affiliation:
Department of Mathematics, University of Florida, Gainesville, Florida 32603
Email:
mjury@math.ufl.edu
DOI:
https://doi.org/10.1090/S0002-9939-07-08931-9
Received by editor(s):
July 27, 2006
Received by editor(s) in revised form:
September 19, 2006
Published electronically:
August 15, 2007
Communicated by:
Joseph A. Ball
Article copyright:
© Copyright 2007
American Mathematical Society
The copyright for this article reverts to public domain 28 years after publication.