Remote Access Proceedings of the American Mathematical Society
Green Open Access

Proceedings of the American Mathematical Society

ISSN 1088-6826(online) ISSN 0002-9939(print)

 
 

 

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

Abstract | References | Similar Articles | Additional Information

Abstract: We prove that the norm of a weighted composition operator on the Hardy space $ H^2$ 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 $ H^2$ 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.


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

  • 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 (2003b:47001)
  • 2. Joseph A. Cima, Charles S. Stanton, and Warren R. Wogen, On boundedness of composition operators on $ H\sp{2}(B\sb{2})$, Proc. Amer. Math. Soc. 91 (1984), no. 2, 217-222. MR 740174 (85j:47030)
  • 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 (2002j:47045)
  • 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 (97i:47056)
  • 5. -, Linear fractional maps of the ball and their composition operators, Acta Sci. Math. (Szeged) 66 (2000), no. 1-2, 351-376. MR 1768872 (2001g:47041)
  • 6. Louis de Branges and James Rovnyak, Square summable power series, Holt, Rinehart and Winston, New York, 1966. MR 0215065 (35:5909)
  • 7. Barbara D. MacCluer, Compact composition operators on $ H\sp p(B\sb N)$, Michigan Math. J. 32 (1985), no. 2, 237-248. MR 783578 (86g:47037)
  • 8. Eric A. Nordgren, Composition operators, Canad. J. Math. 20 (1968), 442-449. MR 0223914 (36:6961)
  • 9. Donald Sarason, Sub-Hardy Hilbert spaces in the unit disk, University of Arkansas Lecture Notes in the Mathematical Sciences, 10, John Wiley & Sons Inc., New York, 1994, A Wiley-Interscience Publication. MR 1289670 (96k:46039)
  • 10. Serguei Shimorin, Commutant lifting and factorization of reproducing kernels, J. Funct. Anal. 224 (2005), no. 1, 134-159. MR 2139107 (2005m:47043)

Similar Articles

Retrieve articles in Proceedings of the American Mathematical Society with MSC (2000): 47B33, 47B32, 46E22

Retrieve articles in all journals with MSC (2000): 47B33, 47B32, 46E22


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.

American Mathematical Society