Remote Access Transactions of the American Mathematical Society
Green Open Access

Transactions of the American Mathematical Society

ISSN 1088-6850(online) ISSN 0002-9947(print)

 
 

 

Hypercyclic operators that commute with the Bergman backward shift


Authors: Paul S. Bourdon and Joel H. Shapiro
Journal: Trans. Amer. Math. Soc. 352 (2000), 5293-5316
MSC (2000): Primary 47B38
DOI: https://doi.org/10.1090/S0002-9947-00-02648-9
Published electronically: July 18, 2000
MathSciNet review: 1778507
Full-text PDF Free Access

Abstract | References | Similar Articles | Additional Information

Abstract:

The backward shift $B$ on the Bergman space of the unit disc is known to be hypercyclic (meaning: it has a dense orbit). Here we ask: ``Which operators that commute with $B$ inherit its hypercyclicity?'' We show that the problem reduces to the study of operators of the form $\varphi(B)$ where $\varphi$ is a holomorphic self-map of the unit disc that multiplies the Dirichlet space into itself, and that the question of hypercyclicity for such an operator depends on how freely $\varphi(z)$ is allowed to approach the unit circle as $\vert z\vert\to 1-$.


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

  • 1. A. B. Aleksandrov, A. É. Dzrbasjan, and V. P. Havin, On the Carleson formula for the Dirichlet integral of an analytic function, (Russian), Vestnik Leningrad. University. Mat. Mekh. Astronom 4 (1979), 8-14. English translation: Vestnik Leningrad Univ. Math. 12 (1980), 237-245. MR 81d:30051
  • 2. S. Axler and A. L. Shields, Univalent multipliers of the Dirichlet space, Michigan Math. J. 32 (1985), 65-80. MR 86c:30043
  • 3. S. R. Bell, The Cauchy Transform, Potential Theory, and Conformal Mapping, CRC Press, Boca Raton, 1992. MR 94k:30013
  • 4. S. R. Bell and S. G. Krantz, Smoothness to the boundary of conformal maps, Rocky Mountain J. Math. 17 (1987) 24-40. MR 88e:30023
  • 5. P. S. Bourdon, J. A. Cima and A. L. Matheson, Compact composition operators on BMOA, Trans. Amer. Math. Soc. 351 (1999), 2183-2196. MR 99i:47052
  • 6. L. Brown and A. L. Shields, Cyclic vectors in the Dirichlet space, Trans. Amer. Math. Soc. 285 (1984), 269-303. MR 86d:30079
  • 7. L. Carleson, A representation formula for the Dirichlet integral, Math. Z. 73 (1960), 190-196. MR 22:3803
  • 8. L. Carleson, Sets of uniqueness for functions regular in the unit circle, Acta Math. 87 (1952), 325-345. MR 14:261a
  • 9. W. G. Cochran, J. H. Shapiro and D. C. Ullrich, Random Dirichlet functions: multipliers and smoothness, Canadian J. Math. 45 (2) (1993), 255-268. MR 94f:30001
  • 10. C. C. Cowen and B. D. MacCluer, Composition Operators on Spaces of Analytic Functions, CRC Press, Boca Raton, 1995. MR 97i:47056
  • 11. P. L. Duren, Theory of $H^p$ Spaces, Pure Appl. Math., 38, Academic Press, 1970. MR 42:3552
  • 12. C. Kitai, Invariant closed sets for linear operators, Thesis, Univ. of Toronto, 1982.
  • 13. E. Flytzanis, Unimodular eigenvalues and linear chaos in Hilbert spaces, Geometric and Functional Analysis 5 (1995), 1-13. MR 95k:28034
  • 14. L. Gårding and L. Hörmander, Strongly subharmonic functions, Math. Scand. 15 (1964), 93-96. MR 31:3621
  • 15. R. M. Gethner and J. H. Shapiro, Universal vectors for operators on spaces of holomorphic functions, Proc. Amer. Math. Soc. 100 (1987), 281-288. MR 88g:47060
  • 16. G. Godefroy and J. H. Shapiro, Operators with dense, invariant, cyclic vector manifolds, J. Funct. Anal. 98 (1991), 229-269. MR 92d:47029
  • 17. K.-G. Grosse-Erdmann, Universal families and hypercyclic operators, Bull. Amer. Math. Soc. 36 (1999) 345-381. MR 2000c:47001
  • 18. P. R. Halmos, A Hilbert Space Problem Book, Second Edition, Springer-Verlag, 1982. MR 84e:47001
  • 19. G. Herzog and C. Schmoeger, On operators $T$ such that $f(T)$ is hypercyclic, Studia Math. 108 (3) (1994), 209-216. MR 95f:47031
  • 20. Y. Katznelson, An Introduction to Harmonic Analysis, Second ed., Dover, New York, 1976. MR 54:10976
  • 21. D. J. Newman and H. S. Shapiro, The Taylor coefficients of inner functions, Michigan Math. J. 9 (1962) 249-255. MR 26:6371
  • 22. C. Robinson, Dynamical Systems. Stability, Symbolic Dynamics, and Chaos, CRC Press, 1995. MR 97e:58064
  • 23. W. Rudin, Real and Complex Analysis, Third Edition, McGraw-Hill, New York, 1987. MR 88k:00002
  • 24. J. H. Shapiro, The essential norm of a composition operator, Annals of Math. 125 (1987), 375-404. MR 88c:47058
  • 25. J. H. Shapiro, Composition Operators and Classical Function Theory, Springer-Verlag, 1993. MR 94k:47049
  • 26. J. H. Shapiro and C. Sundberg, Isolation amongst the composition operators, Pacific J. Math. 145 (1990), 117-152. MR 92g:47041
  • 27. A. L. Shields, Weighted shift operators and analytic function theory, Math. Surveys 13: Topics in Operator Theory, C. Pearcy, ed., American Math. Society 1974, 49-128. MR 50:14341
  • 28. D. A. Stegenga, Bounded Toeplitz operators on $H^1$and applications of the duality between $H^1$ and the functions of bounded mean oscillation, American J. Math. 98 (1973), 573-589. MR 54:8340
  • 29. D. A. Stegenga, Multipliers of the Dirichlet space, Illinois J. Math 24 (1980), 113-139. MR 81a:30027
  • 30. G. D. Taylor, Multipliers on $D_\alpha$, Trans. Amer. Math. Soc. 123 (1966), 229-240. MR 34:6514

Similar Articles

Retrieve articles in Transactions of the American Mathematical Society with MSC (2000): 47B38

Retrieve articles in all journals with MSC (2000): 47B38


Additional Information

Paul S. Bourdon
Affiliation: Department of Mathematics, Washington and Lee University, Lexington, Virginia 24450
Email: pbourdon@wlu.edu

Joel H. Shapiro
Affiliation: Department of Mathematics, Michigan State University, East Lansing, Michigan 48824-1027
Email: shapiro@math.msu.edu

DOI: https://doi.org/10.1090/S0002-9947-00-02648-9
Keywords: Hypercyclic operator, Bergman space, backward shift
Received by editor(s): January 28, 1999
Received by editor(s) in revised form: September 13, 1999
Published electronically: July 18, 2000
Additional Notes: Both authors were supported in part by the National Science Foundation
Article copyright: © Copyright 2000 American Mathematical Society

American Mathematical Society