Advertisement

On Nontangential Limits and Shift Invariant Subspaces

  • John R. AkeroydEmail author
  • John B. Conway
  • Liming Yang
Article
  • 19 Downloads

Abstract

In 1998, Conway and Yang wrote a paper (Holomorphic spaces, MSRI Publications, vol 33, pp 201–209, 1998) in which they posed a number of open questions regarding the shift on \(P^t(\mu )\) spaces. A few of these have been completely resolved, while at least one remains wide open. In this paper, we review some of the solutions, mention some alternate approaches and discuss further the problem that remains unsolved.

Keywords

Nontangential limits Shift invariant subspaces Bounded point evaluations 

Mathematics Subject Classification

Primary 47A15 Secondary 30C85 31A15 46E15 47B38 

Notes

References

  1. 1.
    Akeroyd, J.R.: Another look at some index theorems for the shift. Indiana Univ. Math. J. 50, 705–718 (2001)MathSciNetCrossRefGoogle Scholar
  2. 2.
    Akeroyd, J.R.: A note concerning the index of the shift. Proc. Am. Math. Soc. 130, 3349–3354 (2002)MathSciNetCrossRefGoogle Scholar
  3. 3.
    Aleman, A., Richter, S., Sundberg, C.: Beurling’s theorem for the Bergman space. Acta Math. 177(2), 275–310 (1996)MathSciNetCrossRefGoogle Scholar
  4. 4.
    Aleman, A., Richter, S., Sundberg, C.: Nontangential limits in \(P^t(\mu )\)-spaces and the index of invariant subspaces. Ann. Math. 169(2), 449–490 (2009)MathSciNetCrossRefGoogle Scholar
  5. 5.
    Aleman, A., Richter, S., Sundberg, C.: A quantitative estimate for bounded point evaluations in \(P^t(\mu )\)-spaces. In: Topics in Operator Theory. Operators, Matrices and Analytic Functions. Operator Theory: Advances and Applications, vol. 202, pp. 1–10. Birkhuser, Basel (2010)Google Scholar
  6. 6.
    Brennan, J.E.: The structure of certain spaces of analytic functions. Comput. Methods Funct. Theory 8(2), 625–640 (2008)MathSciNetCrossRefGoogle Scholar
  7. 7.
    Browder, A.: Point derivations on function algebras. J. Funct. Anal. 1, 22–27 (1967)MathSciNetCrossRefGoogle Scholar
  8. 8.
    Conway, J.B.: Subnormal Operators. Pitman Publishing Inc., London (1981)zbMATHGoogle Scholar
  9. 9.
    Conway, J.B.: The theory of subnormal operators. In: Mathematical Survey and Monographs, vol. 36 (1991)Google Scholar
  10. 10.
    Conway, J.B., Elias, N.: Analytic bounded point evaluations for spaces of rational functions. J. Funct. Anal. 117, 1–24 (1993)MathSciNetCrossRefGoogle Scholar
  11. 11.
    Conway, J.B., Yang, L.: Some open problems in the theory of subnormal operators. In: Holomorphic Spaces, MSRI Publications, vol 33, pp. 201–209 (1998)Google Scholar
  12. 12.
    Gamelin, T.W.: Uniform algebras. American Mathematical Society, Rhode Island (1969)zbMATHGoogle Scholar
  13. 13.
    Hedenmalm, H., Zhu, K.: On the failure of optimal factorization for certain weighted Bergman spaces. Complex Var. 19, 165–176 (1992)MathSciNetzbMATHGoogle Scholar
  14. 14.
    Olin, R.F., Thomson, J.E.: Some index theorems for subnormal operators. J. Oper. Theory 3, 115–142 (1980)MathSciNetzbMATHGoogle Scholar
  15. 15.
    Shimorin, S.: Wold-type decompositions and wandering subspaces for operators close to isometries. J. Reine Angew. Math. 531, 147–189 (2001)MathSciNetzbMATHGoogle Scholar
  16. 16.
    Thomson, J.E.: Approximation in the mean by polynomials. Ann. Math. 133(3), 477–507 (1991)MathSciNetCrossRefGoogle Scholar
  17. 17.
    Tolsa, X.: On the analytic capacity \(\gamma +\). Indiana Univ. Math. J. 51(2), 317–343 (2002)MathSciNetCrossRefGoogle Scholar
  18. 18.
    Tolsa, X.: Painleves problem and the semiadditivity of analytic capacity. Acta Math. 51(1), 105–149 (2003)MathSciNetCrossRefGoogle Scholar
  19. 19.
    Tolsa, X.: Analytic Capacity, the Cauchy Transform, and Non-homogeneous Calderon–Zygmund Theory. Birkhauser/Springer, Cham (2014)CrossRefGoogle Scholar
  20. 20.
    Yang, L.: Bounded point evaluations for rationally multicyclic subnormal operators. J. Math. Anal. Appl. 458, 1059–1072 (2018)MathSciNetCrossRefGoogle Scholar

Copyright information

© Springer Nature Switzerland AG 2019

Authors and Affiliations

  • John R. Akeroyd
    • 1
    Email author
  • John B. Conway
    • 2
  • Liming Yang
    • 3
  1. 1.Department of MathematicsUniversity of ArkansasFayettevilleUSA
  2. 2.Department of MathematicsThe George Washington UniversityWashingtonUSA
  3. 3.Department of MathematicsVirginia Polytechnic and State UniversityBlacksburgUSA

Personalised recommendations