Skip to main content
SpringerLink
Log in

Corona Problem for H ∞ on Riemann Surfaces

  • Chapter
  • First Online:
The Corona Problem

Part of the book series: Fields Institute Communications ((FIC,volume 72))

  • 1059 Accesses

Abstract

In this paper we survey some results and methods related to the famous corona problem for algebras H ∞ of bounded holomorphic functions on Caratheodory hyperbolic Riemann surfaces.

This is a preview of subscription content, log in via an institution to check access.

Access this chapter

Log in via an institution
eBook
USD 16.99
Price excludes VAT (USA)
  • Available as EPUB and PDF
  • Read on any device
  • Instant download
  • Own it forever
Softcover Book
USD 54.99
Price excludes VAT (USA)
  • Compact, lightweight edition
  • Dispatched in 3 to 5 business days
  • Free shipping worldwide - see info
Hardcover Book
USD 54.99
Price excludes VAT (USA)
  • Durable hardcover edition
  • Dispatched in 3 to 5 business days
  • Free shipping worldwide - see info

Tax calculation will be finalised at checkout

Purchases are for personal use only

Institutional subscriptions

References

  1. N. L. Alling, A proof of the corona conjecture for finite open Riemann surfaces. Bull. Amer. Math. Soc. 70 (1964), 110–112.

    Article  MATH  MathSciNet  Google Scholar 

  2. N. L. Alling, Extensions of meromorphic function rings over non-compact Riemann surfaces, I. Math. Z. 89 (1965), 273–299.

    Article  MathSciNet  Google Scholar 

  3. D. E. Barrett, and J. Diller, A new construction of Riemann surfaces with corona. J. Geom. Anal. 8 (1998), 341–347.

    Article  MATH  MathSciNet  Google Scholar 

  4. M. Behrens, On the corona problem for a class of infinitely connected domains. Bull. Amer. Math. Soc. 76 (1970), 387–391.

    Article  MATH  MathSciNet  Google Scholar 

  5. M. Behrens, The maximal ideal space of algebras of bounded analytic functions on infinitely connected domains. Trans. Amer. Math. Soc. 161 (1971), 358–380.

    Article  MathSciNet  Google Scholar 

  6. A. Beurling, On two problems concerning linear transformations in Hilbert space. Acta Math. 81 (1948), 17.

    MathSciNet  Google Scholar 

  7. J. Bourgain and O. Reinov, On the approximation properties for the space H ∞. Math. Nachr. 122 (1983) 19–27.

    Article  MathSciNet  Google Scholar 

  8. A. Brudnyi, Matrix-valued corona theorem for multiply connected domains. Indiana Univ. Math. J. 49 (4) (2000), 1405–1410.

    Article  MATH  MathSciNet  Google Scholar 

  9. A. Brudnyi, Topology of the maximal ideal space of H ∞. J. Funct. Anal. 189 (2002), 21–52.

    Article  MATH  MathSciNet  Google Scholar 

  10. A. Brudnyi, A uniqueness property for H ∞ on coverings of projective manifolds. Michigan Math. J. 51 (2003), no. 3, 503–507.

    Article  MATH  MathSciNet  Google Scholar 

  11. A. Brudnyi, Projections in the space H ∞ and the Corona Theorem for coverings of bordered Riemann surfaces. Ark. Mat. 42 (2004), no. 1, 31–59.

    MATH  MathSciNet  Google Scholar 

  12. A. Brudnyi, Grauert and Lax-Halmos type theorems and extension of matrices with entries in H ∞. J. Funct. Anal. 206 (2004), 87–108.

    Article  MATH  MathSciNet  Google Scholar 

  13. A. Brudnyi, Corona Theorem for H ∞ on Coverings of Riemann Surfaces of Finite Type. Michigan Math. J. 56 (2008), 283–299.

    Article  MATH  MathSciNet  Google Scholar 

  14. A. Brudnyi, Extension of matrices with entries in H ∞ on coverings of Riemann surfaces of finite type. St. Petersburg J. Math. 21 (2009), no. 3, 79–92.

    MathSciNet  Google Scholar 

  15. A. Brudnyi, Holomorphic Banach vector bundles on the maximal ideal space of H ∞ and the operator corona problem of Sz.-Nagy. Adv. Math. 232 (2013), 121–141.

    Google Scholar 

  16. A. Brudnyi, Stein-like theory for Banach-valued holomorphic functions on the maximal ideal space of H ∞. Invent. math. 193 (2013), 187–227.

    Article  MATH  MathSciNet  Google Scholar 

  17. A. Brudnyi, On the completion problem for algebra H ∞, J. Func. Anal. 266 (2014), 4293–4313.

    Article  MATH  MathSciNet  Google Scholar 

  18. A. Brudnyi, and A. Sasane, Sufficient conditions for the projective freeness of Banach algebras. J. Funct. Anal. 257 (2009), 4003–4014.

    Article  MATH  MathSciNet  Google Scholar 

  19. L. Bungart, On analytic fibre bundles I. Holomorphic fibre bundles with infinite dimensional fibres. Topology 7 (1968), 55–68.

    MATH  MathSciNet  Google Scholar 

  20. L. Carleson, Interpolation of bounded analytic functions and the corona problem. Ann. of Math. 76 (1962), 547–559.

    Article  MATH  MathSciNet  Google Scholar 

  21. L. Carleson, On H ∞ in multiply connected domains. In Conference on Harmonic Analysis in Honor of Antoni Zygmund, Vol. II (Beckman, W., Calderón, A. P., Fefferman, R. and Jones. P. W., eds.), pp. 349–372, Wadsworth, Belmont, Calif., 1983.

    Google Scholar 

  22. P.M. Cohn, Free rings and their relations. Second edition. London Mathematical Society Monographs 19, Academic Press, Inc. [Harcourt Brace Jovanovich, Publishers], London, 1985.

    Google Scholar 

  23. P.M. Cohn, Some remarks on projective free rings. Dedicated to the memory of Gian-Carlo Rota. Algebra Universalis, no.2, 49 (2003), 159–164.

    Google Scholar 

  24. W. Deeb, A class of infinitely connected domain and the corona. Trans. Amer. Math. Soc. 31 (1977), 101–106.

    Article  MathSciNet  Google Scholar 

  25. W. Deeb and D.R. Wilken, Δ domains and the corona.Trans. Amer. Math. Soc. 231 (1977), 107–115.

    MATH  MathSciNet  Google Scholar 

  26. C. J. Earle and A. Marden, Projections to automorphic functions. Proc. Amer. Math. Soc. 19 (1968), 274–278.

    Article  MATH  MathSciNet  Google Scholar 

  27. C. J. Earle and A. Marden, On Poincaré series with application to H p spaces on bordered Riemann surfaces. Illinois J. Math. 13 (1969), 202–219.

    MATH  MathSciNet  Google Scholar 

  28. F. Forelli, Bounded holomorphic functions and projections. Illinois J. Math. 10 (1966), 367–380.

    MATH  MathSciNet  Google Scholar 

  29. P. A. Fuhrmann, On the corona theorem and its application to spectral problems in Hilbert space. Trans. Amer. Math. Soc. 132 (1968), 55–66.

    Article  MATH  MathSciNet  Google Scholar 

  30. T. W. Gamelin, Uniform algebras and Jensen measures. London Math. Soc. Lecture Notes Series 32. Cambridge Univ. Press, Cambridge-New York, 1978.

    Google Scholar 

  31. T. W. Gamelin, Localization of the corona problem. Pacific J. Math. 34 (1970), 73–81.

    Article  MATH  MathSciNet  Google Scholar 

  32. J. B. Garnett, Bounded analytic functions. Academic Press, 1981.

    Google Scholar 

  33. J. B. Garnett and P. W. Jones, The corona theorem for Denjoy domains. Acta Math. 155 (1985), 27–40.

    Article  MATH  MathSciNet  Google Scholar 

  34. J. B. Garnett and A. Nicolau, Interpolating Blaschke products generate H ∞. Pacific J. Math. 173 no. 2 (1996), 501–510.

    MATH  MathSciNet  Google Scholar 

  35. H. Grauert, Analytische Faserungen über holomorph-vollständigen Räumen. Math. Ann. 135 (1958), 263–273.

    Article  MATH  MathSciNet  Google Scholar 

  36. J. Handy, The corona theorem on the complement of certain square Cantor sets. J. Anal. Math. 108 (1) (2009), 1–18.

    Article  MATH  MathSciNet  Google Scholar 

  37. M. Hara and M. Nakay, Corona theorem with bounds for finitely sheeted disks. Tohoku Math. J. 37 (1985), 225–240.

    Article  MathSciNet  Google Scholar 

  38. M. Hasumi, Hardy classes on infinitely connected Riemann surfaces. Lecture Notes in Math. 1027, Springer-Verlag, Berlin Heidelberg, 1983.

    Google Scholar 

  39. K. Hoffman, Bounded analytic functions and Gleason parts. Ann. of Math. 86 (1967), 74–111.

    Article  MATH  MathSciNet  Google Scholar 

  40. L. Hörmander, Generators for some rings of analytic functions. Bull. Amer. Math. Soc. 73 (1967), 943–949.

    Article  MATH  MathSciNet  Google Scholar 

  41. P. W. Jones and D. Marshall, Critical points of Green’s functions, harmonic measure and the corona theorem. Ark. Mat. 23 (1985), 281–314.

    Article  MATH  MathSciNet  Google Scholar 

  42. T.Y. Lam, Serre’s conjecture. Lecture Notes in Mathematics 635, Springer-Verlag, Berlin-New York, 1978.

    Google Scholar 

  43. F. Lárusson, Holomorphic functions of slow growth on nested covering spaces of compact manifolds. Canad. J. Math. 52 (2000), 982–998.

    Article  MATH  MathSciNet  Google Scholar 

  44. P. D. Lax, Translation invariant spaces. Acta Math. 101 (1959), 163–178.

    Article  MATH  MathSciNet  Google Scholar 

  45. V. Lin, Holomorphic fibering and multivalued functions of elements of a Banach algebra. Funct. Anal. and its Appl. English translation, 7 (2) (1973), 122–128.

    Article  MATH  Google Scholar 

  46. B. S. Mityagin, The homotopy structure of the linear group of a Banach space. Russ. Math. Surv. 25 (5) (1970), 59–103.

    Article  MATH  MathSciNet  Google Scholar 

  47. C. N. Moore, The corona theorem for domains whose boundary lies in a smooth curve. Proc. Amer. Math. Soc. 100 (1987), no. 2, 266–270.

    Article  MATH  MathSciNet  Google Scholar 

  48. K. Nagami, Dimension theory. Academic Press. New York, 1970.

    MATH  Google Scholar 

  49. B. Sz.-Nagy, A problem on operator valued bounded analytic functions. Zap. Nauchn. Sem. LOMI, 81(1978), 99.

    Google Scholar 

  50. B. Sz.-Nagy and C. Foias, Harmonic analysis of operators on Hilbert space. Translated from the French and revised. North-Holland Publishing Co., Amsterdam, 1970.

    Google Scholar 

  51. M. Nakai, Corona problem for Riemann surfaces of Parreau-Widom type. Pacific J. Math. 103 (1982), 103–109.

    Article  MATH  MathSciNet  Google Scholar 

  52. M. Nakay, The corona problem on finitely sheeted covering surfaces. Nagoya Math. J. 92 (1983), 163–173.

    MathSciNet  Google Scholar 

  53. J. Narita, A remark on the corona problem for plane domains. J. Math. Kyoto Univ. 25 (1985), 293–298.

    MATH  MathSciNet  Google Scholar 

  54. D. J. Newman, Interpolation in \(H^{\infty }(\mathbb{D})\). Trans. Amer. Math. Soc. 92 no. 2 (1959), 502–505.

    Google Scholar 

  55. B. M. NewDelman, Homogeneous subsets of a Lipschitz graph and the Corona theorem. Publ. Mat. 55 (2011), 93–121.

    Article  MATH  MathSciNet  Google Scholar 

  56. N. K. Nikolski, Operators, functions, and systems: an easy reading. Vol. 1: Hardy, Hankel, and Toeplitz. Mathematical Surveys and Monographs, vol. 92, American Mathematical Society, Providence, RI, 2002, Translated from the French by Andreas Hartmann.

    Google Scholar 

  57. N. K. Nikolski, Treatise on the shift operator. Grundlehren der Mathematischen Wissenschaften [Fundamental Principles of Mathematical Sciences], vol. 273, Springer–Verlag, Berlin, 1986, Spectral function theory, With an appendix by S. V. Hruščev [S. V. Khrushchëv] and V. V. Peller, Translated from the Russian by Jaak Peetre.

    Google Scholar 

  58. M. E. Novodvorskii, On some homotopic invariants of the maximal ideals space. Math. Notes 1 (1967), 487–494. (Russian)

    Article  MathSciNet  Google Scholar 

  59. B. Oh, An explicit example of Riemann surfaces with large bound on corona solutions. Pacific J. Math. 228 (2006), 297–304.

    Article  MATH  MathSciNet  Google Scholar 

  60. B. Oh, A short proof of Hara and Nakai’s theorem. Proc. Amer. Math. Soc. 136 (2008), 4385–4388.

    Article  MATH  MathSciNet  Google Scholar 

  61. Ch. Pommerenke, On the Green’s function of Fuchsian groups. Ann. Acad. Sci. Fenn. 2 (1976), 409–427.

    Article  MATH  MathSciNet  Google Scholar 

  62. A. Quadrat, The fractional representation approach to synthesis problems: an algebraic analysis viewpoint. II. Internal stabilization. SIAM J. Control Optim. 42 no. 1 (2003), 300–320.

    Article  MATH  MathSciNet  Google Scholar 

  63. M. Rosenblum, A corona theorem for countably many functions. Integral Equations Operator Theory 3 (1980), 125–137.

    Article  MATH  MathSciNet  Google Scholar 

  64. S. Scheinberg, Thesis. Princeton University, 1962.

    Google Scholar 

  65. E. L. Stout, Two theorems concerning functions holomorphic on multiply connected domains. Bull. Amer. Math. Soc. 69 (1963), 527–530.

    Article  MATH  MathSciNet  Google Scholar 

  66. E. L. Stout, Bounded holomorphic functions on finite Riemann surfaces. Trans. Amer. Math. Soc. 120 (1965), 255–285.

    MATH  MathSciNet  Google Scholar 

  67. E. L. Stout, On some algebras of analytic function on finite open Riemann surfaces. Math. Z. 92 (1966), 366–379. Corrections to: On some algebras of analytic function on finite open Riemann surfaces. Math. Z. 95 (1967), 403–404.

    Google Scholar 

  68. D. Suárez, Čech cohomology and covering dimension for the H ∞ maximal ideal space. J. Funct. Anal. 123 (1994), 233–263.

    Article  MATH  MathSciNet  Google Scholar 

  69. D. Suárez, Trivial Gleason parts and the topological stable rank of H ∞. Amer. J. Math. 118 (1996), 879–904.

    Article  MATH  MathSciNet  Google Scholar 

  70. J. L. Taylor, Topological invariants of the maximal ideal space of a Banach algebra. Adv. Math. 19 (1976), 149–206.

    Article  MATH  Google Scholar 

  71. V. Tolokonnikov, Estimates for Carleson Corona theorem, ideals of algebra H ∞, Sekefalvi-Nagy problem. Zap. Nauch. Sem. Leningrad. Otdel. Mat. Inst. Steklov. (LOMI) 113 (1981), 178–190. (Russian)

    MATH  MathSciNet  Google Scholar 

  72. V. Tolokonnikov, Extension problem to an invertible matrix. Proc. Amer. Math. Soc. 117 (1993), no. 4, 1023–1030.

    Article  MATH  MathSciNet  Google Scholar 

  73. S. Treil, Angles between co-invariant subspaces, and the operator corona problem. The Szökefalvi-Nagy problem. Dokl. Akad. Nauk SSSR 302 (1988), 1063–1068. (Russian)

    Google Scholar 

  74. S. R. Treil, Geometric methods in spectral theory of vector-valued functions: some recent results, Toeplitz operators and spectral function theory. Oper. Theory Adv. Appl., vol. 42, Birkhäuser, Basel, 1989, pp. 209–280.

    Google Scholar 

  75. S. Treil, Geometric methods in spectral theory of vector-valued functions: Some recent results. In Toeplitz Operators and Spectral Function Theory, Oper. Theory Adv. Appl., vol. 42, Birkhäuser Verlag, Basel 1989, pp. 209–280.

    Google Scholar 

  76. S. Treil, The stable rank of H ∞ equals 1. J. Funct. Anal. 109 (1992), 130–154.

    Article  MATH  MathSciNet  Google Scholar 

  77. S. R. Treil, Unconditional bases of invariant subspaces of a contraction with finite defects. Indiana Univ. Math. J. 46 (1997), no. 4, 1021–1054.

    Article  MATH  MathSciNet  Google Scholar 

  78. S. Treil, Lower bounds in the matrix corona theorem and the codimension one conjecture. GAFA 14 5 (2004), 1118–1133.

    MATH  MathSciNet  Google Scholar 

  79. S. Treil, An operator Corona theorem. Indiana Univ. Math. J. 53 (2004), no. 6, 1763–1780.

    Article  MATH  MathSciNet  Google Scholar 

  80. S. Treil and B. Wick, Analytic projections, Corona Problem and geometry of holomorphic vector bundles. J. Amer. Math. Soc. 22 (2009), no. 1, 55–76.

    MATH  MathSciNet  Google Scholar 

  81. A. Uchiyama, Corona theorems for countably many functions and estimates for their solutions, (1980) (preprint).

    Google Scholar 

  82. M. Vidyasagar, Control system synthesis: a factorization approach. MIT Press Series in Signal Processing, Optimization, and Control 7, MIT Press, Cambridge, MA, 1985.

    Google Scholar 

  83. H. Widom, H p sections of vector bundles over Riemann surfaces. Ann. of Math. 94 (1971), 304–324.

    Article  MATH  MathSciNet  Google Scholar 

Download references

Acknowledgements

Research supported in part by NSERC. I thank the anonymous referee for useful remarks improving the presentation of the paper.

Author information

Authors and Affiliations

  1. Department of Mathematics and Statistics, University of Calgary, Calgary, AB, Canada, T2N 1N4

    Alexander Brudnyi

Authors
  1. Alexander Brudnyi
    View author publications

    You can also search for this author in PubMed Google Scholar

Corresponding author

Correspondence to Alexander Brudnyi .

Editor information

Editors and Affiliations

  1. Department of Mathematics, Texas A&M University, College Station, Texas, USA

    Ronald G. Douglas

  2. Department of Mathematics, Washington University, St. Louis, Missouri, USA

    Steven G. Krantz

  3. Department of Mathematics and Statistics, McMaster University, Hamilton, Ontario, Canada

    Eric T. Sawyer

  4. Department of Mathematics, Brown University, Providence, Rhode Island, USA

    Sergei Treil

  5. Department of Mathematics, Georgia Institute of Technology, Atlanta, Georgia, USA

    Brett D. Wick

Rights and permissions

Reprints and permissions

Copyright information

© 2014 Springer Science+Business Media New York

About this chapter

Cite this chapter

Brudnyi, A. (2014). Corona Problem for H ∞ on Riemann Surfaces. In: Douglas, R., Krantz, S., Sawyer, E., Treil, S., Wick, B. (eds) The Corona Problem. Fields Institute Communications, vol 72. Springer, New York, NY. https://doi.org/10.1007/978-1-4939-1255-1_2

Download citation

Publish with us

Policies and ethics

Search

Navigation