IMPORTANT NOTICE

The AMS website will be down for maintenance on May 23 between 6:00am - 8:00am EDT. For questions please contact AMS Customer Service at cust-serv@ams.org or (800) 321-4267 (U.S. & Canada), (401) 455-4000 (Worldwide).

 

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)

 
 

 

Similarity to a contraction and hypercontractivity of composition operators


Author: Nizar Jaoua
Journal: Proc. Amer. Math. Soc. 129 (2001), 2085-2092
MSC (1991): Primary 47B38, 47B65
DOI: https://doi.org/10.1090/S0002-9939-00-05843-3
Published electronically: December 28, 2000
MathSciNet review: 1825921
Full-text PDF

Abstract | References | Similar Articles | Additional Information

Abstract: On the Hardy spaces $H^p$ with $1 \leq p<\infty$, we consider the composition operators induced by analytic self-maps of the open unit disc $D$. First, we characterize those which are similar to contractions. Then, we give some necessary and sufficient conditions for them to be hypercontractive. Finally, we prove that, among those ones, only the zero-symbol composition operator sends $H^p$ into $H^{\infty}$ with a norm less than or equal to $1$.


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

  • 1. J. B. Conway, A course in Functional Analysis, Second Edition. Springer-Verlag New York (1990). MR 91e:46001
  • 2. C. C. Cowen and B. D. MacCluer, Composition Operators on Spaces of Analytic Functions, CRC Press, Boca Raton (1995). MR 97i:47056
  • 3. A. Denjoy, Sur l'itération des fonctions analytiques, C. R. Acad. Sci. Paris Sér. A., 182 (1926) 255-257.
  • 4. N. Dunford and J. T. Schwartz, Linear Operators, II (1963). MR 32:6181
  • 5. P. L. Duren, Theory of $H^p$ Spaces, Academic Press (1970). MR 42:3552
  • 6. G. Hoever, Two classroom proofs concerning composition operators, Integr. Equ. Oper. Theory, 27 (1997) 493-496. MR 98b:47041
  • 7. H. Hunziker and H. Jarchow, Composition operators which improve integrability, Math. Nachr., 152 (1991) 83-99. MR 93d:47061
  • 8. N. Jaoua, Propriétés de similarité et de contractivité de certains opérateurs de composition sur des classes de fonctions anlytiques, Thèse, Université de Lille (1999).
  • 9. H. Kamowitz, The spectra of composition operators on $H^p$, J. Funct. Analysis, 18 (1975) 132-150. MR 53:11417
  • 10. E. Nordgren, Composition operators, Canadian J. Math., 20 (1968) 442-449. MR 36:6961
  • 11. -, Composition operators on Hilbert spaces, Lecture Notes in Math., 693 (1977), 37-63. MR 80d:47046
  • 12. G. Pisier, Similarity problems and completely bounded maps, Springer Lecture Notes, 1618 (1996). MR 98d:47002
  • 13. -, A polynomially bounded operator on Hilbert space which is not similar to a contraction, J. Amer. Math. Soc., 10, 2 (1997) 351-369. MR 97f:47002
  • 14. H. J. Schwartz, Composition operators on $H^p$, Thesis, University of Toledo (1969).
  • 15. F. B. Weissler, Logarithmic Sobolev inequalities and hypercontracive estimates on the circle, J. Funct. Analysis, 37 (1980) 218-234. MR 81k:42007
  • 16. J. Wolff, Sur l'itération des fonctions analytiques, C. R. Acad. Sci. Paris Sér. A. 182 (1926) 42-43.
  • 17. K. Zhu, Operator Theory in Function Spaces, Marcel Dekker, New York, (1990). MR 92c:47031

Similar Articles

Retrieve articles in Proceedings of the American Mathematical Society with MSC (1991): 47B38, 47B65

Retrieve articles in all journals with MSC (1991): 47B38, 47B65


Additional Information

Nizar Jaoua
Affiliation: Department of Mathematics, Université Lille I, Cité Scientifique, F-59655 Villeneuve d’Ascq, France
Email: jaoua@agat.univ-lille1.fr

DOI: https://doi.org/10.1090/S0002-9939-00-05843-3
Keywords: Composition operators, Hardy spaces, contraction, hypercontractive (or $\beta$-contractive) operator
Received by editor(s): March 10, 1999
Received by editor(s) in revised form: November 17, 1999
Published electronically: December 28, 2000
Communicated by: David R. Larson
Article copyright: © Copyright 2000 American Mathematical Society

American Mathematical Society