Skip to main content
SpringerLink
Log in

A History of the Corona Problem

  • Chapter
  • First Online:
The Corona Problem

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

Abstract

We give a history of the Corona Problem in both the one variable and the several variable setting. We also describe connections with functional analysis and operator theory. A number of open problems are described.

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

Notes

  1. 1.

    This paper is very entertaining because its author I. J. Schark is a fiction. “I. J. Schark” is actually an acronym for the authors Irving Kaplansky, John Wermer, Shizuo Kakutani, R. Creighton Buck, Halsey Royden, Andrew Gleason, Richard Arens and Kenneth Hoffman. The letters of “I. J. Schark” come from their first initials. The references to this paper, plus consultation with experts, show that virtually no work was done on the Corona Problem between 1941 and 1961.

  2. 2.

    Of course there are some technical details about interpreting the boundary values of the function b j, k , but one can avoid such difficulties by first assuming that the corona data is analytic in a bigger disc and then using a standard normal family argument.

  3. 3.

    That is, a domain supports a (unbounded) holomorphic function that cannot be analytically continued to any larger domain.

References

  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.

    Google Scholar 

  2. Jim Agler and John E., Nevanlinna-Pick interpolation on the bidisk, J. Reine Angew. Math. 506 (1999), 191–204.

    Article  MATH  MathSciNet  Google Scholar 

  3. Norman 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 

  4. É. Amar, On the Toëplitz corona problem, Publ. Mat. 47 (2003), no. 2, 489–496.

    Google Scholar 

  5. É. Amar and Joaquim Bruna, On H p -solutions of the Bezout equation in the ball, Proceedings of the Conference in Honor of Jean-Pierre Kahane (Orsay, 1993), 1995, pp. 7–15 (English, with English and French summaries).

    Google Scholar 

  6. Cǎlin-Grigore Ambrozie and Dan Timotin, On an intertwining lifting theorem for certain reproducing kernel Hilbert spaces, Integral Equations Operator Theory 42 (2002), no. 4, 373–384.

    Article  MATH  MathSciNet  Google Scholar 

  7. Mats Andersson, Topics in complex analysis, Universitext, Springer-Verlag, New York, 1997.

    Google Scholar 

  8. É. Amar and Joaquim Bruna, The H 2 corona problem and \(\bar{\partial }_{b}\) in weakly pseudoconvex domains, Trans. Amer. Math. Soc. 342 (1994), no. 1, 241–255.

    Google Scholar 

  9. É. Amar and Joaquim Bruna, On the H p corona problem, Bull. Sci. Math. 118 (1994), no. 3, 287–306.

    Google Scholar 

  10. Mats Andersson and Hasse Carlsson, Estimates of solutions of the Hp and BMOA corona problem, Math. Ann. 316 (2000), no. 1, 83–102.

    Article  MATH  MathSciNet  Google Scholar 

  11. É. Amar and Joaquim Bruna, H p -estimates of holomorphic division formulas, Pacific J. Math. 173 (1996), no. 2, 307–335.

    Google Scholar 

  12. É. Amar and Joaquim Bruna, Wolff type estimates and the H p corona problem in strictly pseudoconvex domains, Ark. Mat. 32 (1994), no. 2, 255–276.

    Google Scholar 

  13. William Arveson, Interpolation problems in nest algebras, J. Functional Analysis 20 (1975), no. 3, 208–233.

    Article  MATH  MathSciNet  Google Scholar 

  14. Joseph A. Ball, Tavan T. Trent, and Victor Vinnikov, Interpolation and commutant lifting for multipliers on reproducing kernel Hilbert spaces, Operator theory and analysis (Amsterdam, 1997), Oper. Theory Adv. Appl., vol. 122, Birkhäuser, Basel, 2001, pp. 89–138.

    Google Scholar 

  15. G. Berg, Bounded holomorphic functions of several variables, Ark. Mat. 20 (1982), 249–270.

    Article  MATH  MathSciNet  Google Scholar 

  16. Jean Bourgain, On finitely generated closed ideals in H ∞ (D), Ann. Inst. Fourier (Grenoble) 35 (1985), no. 4, 163–174 (English, with French summary).

    Google Scholar 

  17. Alexander Brudnyi, Holomorphic Banach vector bundles on the maximal ideal space of H ∞ (D) and the operator corona problem of Sz.-Nagy, Adv. Math. 232 (2013), no. 1, 121–141.

    Article  MATH  MathSciNet  Google Scholar 

  18. G. Berg, An interpolation problem for bounded analytic functions, Amer. J. Math. 80 (1958), 921–930.

    Article  MathSciNet  Google Scholar 

  19. G. Berg, Interpolations by bounded analytic functions and the corona problem, Ann. of Math. (2) 76 (1962), 547–559.

    Google Scholar 

  20. Urban Cegrell, Generalisations of the corona theorem in the unit disc, Proc. Roy. Irish Acad. Sect. A 94 (1994), no. 1, 25–30.

    MATH  MathSciNet  Google Scholar 

  21. G. Berg, A generalization of the corona theorem in the unit disc, Math. Z. 203 (1990), no. 2, 255–261.

    MathSciNet  Google Scholar 

  22. G. Berg, On ideals generated by bounded analytic functions in the bidisc, Bull. Soc. Math. France 121 (1993), no. 1, 109–116 (English, with English and French summaries).

    Google Scholar 

  23. Joseph A. Cima and Gerald D. Taylor, On the equation f1g1 + f2g2 = 1 in H p, Illinois J. Math. 11 (1967), 431–438.

    MATH  MathSciNet  Google Scholar 

  24. Joseph A. Cima and Gerald D., The Corona Theorem for the Drury-Arveson Hardy space and other holomorphic Besov-Sobolev spaces on the unit ball in \(\mathbb{C}^{n}\), Anal. & PDE 4 (2011), 499–550, available at http://arxiv.org/abs/0811.0627.

  25. Ronald G. Douglas and Jaydeb Sarkar, Some remarks on the Toeplitz corona problem, Hilbert spaces of analytic functions, CRM Proc. Lecture Notes, vol. 51, Amer. Math. Soc., Providence, RI, 2010, pp. 81–89.

    Google Scholar 

  26. John Erik Fornæss and Nessim Sibony, Smooth pseudoconvex domains in C 2 for which the corona theorem and L p estimates for \(\bar{\partial }\) fail, Complex analysis and geometry, Univ. Ser. Math., Plenum, New York, 1993, pp. 209–222.

    Google Scholar 

  27. Paul 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 

  28. Paul A., Localization of the corona problem, Pacific J. Math. 34 (1970), 73–81.

    Article  MathSciNet  Google Scholar 

  29. Theodore Gamelin, Uniform Algebras and Jensen Measures, LMS Lecture Notes Series, vol. 32, Cambridge University Press, Cambridge, 1978.

    Google Scholar 

  30. John B. Garnett, Bounded analytic functions, 1st ed., Graduate Texts in Mathematics, vol. 236, Springer, New York, 2007.

    Google Scholar 

  31. John B. Garnett and Peter W. Jones, The corona theorem for Denjoy domains, Acta Math. 155 (1985), no. 1–2, 27–40.

    Article  MATH  MathSciNet  Google Scholar 

  32. Andrew M. Gleason, Function algebras, Seminar on Analytic Functions 2 (1957), 213–226.

    Google Scholar 

  33. V. P. Havin and N. K. Nikolski (eds.), Linear and complex analysis. Problem book 3. Part I, Lecture Notes in Mathematics, vol. 1573, Springer-Verlag, Berlin, 1994. MR1334345 (96c:00001a)

    Google Scholar 

  34. V. P. Havin and N. K. Nikolski (ed.), Linear and complex analysis. Problem book 3. Part II, Lecture Notes in Mathematics, vol. 1574, Springer-Verlag, Berlin, 1994.

    Google Scholar 

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

    Article  MATH  MathSciNet  Google Scholar 

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

    Article  MATH  MathSciNet  Google Scholar 

  37. M. Jarnicki and P. Pflug, Extension of Holomorphic Functions, Walter de Gruyter & Co., Berlin, 2000.

    Book  MATH  Google Scholar 

  38. Peter W. Jones, Lennart Carleson’s work in analysis, Festschrift in honour of Lennart Carleson and Yngve Domar (Uppsala, 1993), Acta Univ. Upsaliensis Skr. Uppsala Univ. C Organ. Hist., vol. 58, Uppsala Univ., Uppsala, 1995, pp. 17–28.

    Google Scholar 

  39. S. V. Kislyakov and D. V. Rutskiĭ, Some remarks on the corona theorem, Algebra i Analiz 24 (2012), no. 2, 171–191 (Russian, with Russian summary); English transl., St. Petersburg Math. J. 24 (2013), no. 2, 313–326.

    Google Scholar 

  40. Paul Koosis, Introduction to Hp spaces, 2nd ed., Cambridge Tracts in Mathematics, vol. 115, Cambridge University Press, Cambridge, 1998. With two appendices by V. P. Havin [Viktor Petrovich Khavin].

    Google Scholar 

  41. S. V. Kislyakov and D. V. Rutskiĭ, Normed domains of holomorphy, Int. J. Math. Math. Sci. (2010), 18 pp.

    Google Scholar 

  42. S. V. Kislyakov and D. V. Rutskiĭ, Cornerstones of Geometric Function Theory: Explorations in Complex Analysis, 1st ed., Birkhäuser, Boston, 2006.

    Google Scholar 

  43. Kai-Ching Lin, On the H p solutions to the corona equation, Bull. Sci. Math. 118 (1994), no. 3, 271–286.

    MATH  MathSciNet  Google Scholar 

  44. S. V. Kislyakov and D. V. Rutskiĭ, On the constants in the corona theorem and the ideals of H ∞ Houston J. Math. 19 (1993), no. 1, 97–106.

    MathSciNet  Google Scholar 

  45. S. V. Kislyakov and D. V. Rutskiĭ, The corona theorem and interpolating Blaschke products, Indiana Univ. Math. J. 41 (1992), no. 3, 851–859.

    Article  MathSciNet  Google Scholar 

  46. S. V. Kislyakov and D. V. Rutskiĭ, H p-solutions for the corona problem on the polydisc in C n, Bull. Sci. Math. (2) 110 (1986), no. 1, 69–84 (English, with French summary).

    Google Scholar 

  47. D. J. Newman, personal communication.

    Google Scholar 

  48. N. K. Nikol’skiĭ, 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 

  49. J. M. Ortega and J. Fábrega, Corona type decomposition in some Besov spaces, Math. Scand. 78 (1996), no. 1, 93–111.

    MathSciNet  Google Scholar 

  50. J. M. Ortega and J. Fábrega, Pointwise multipliers and corona type decomposition in BMOA, Ann. Inst. Fourier (Grenoble) 46 (1996), no. 1, 111–137 (English, with English and French summaries).

    Google Scholar 

  51. Jordi Pau, On a generalized corona problem on the unit disc, Proc. Amer. Math. Soc. 133 (2005), no. 1, 167–174 (electronic).

    Google Scholar 

  52. Mrinal Raghupathi and Brett D. Wick, Duality, tangential interpolation, and Töoplitz corona problems, Integral Equations Operator Theory 68 (2010), no. 3, 337–355.

    Article  MATH  MathSciNet  Google Scholar 

  53. K. V. Rajeswara Rao, On a generalized corona problem, J. Analyse Math. 18 (1967), 277–278.

    Article  MathSciNet  Google Scholar 

  54. Jean-Pierre Rosay, Une équivalence au corona problem dans C n et un probléme d’idéal dans H ∞(D), J. Functional Analysis 7 (1971), 71–84 (French).

    Google Scholar 

  55. Marvin Rosenblum, A corona theorem for countably many functions, Integral Equations Operator Theory 3 (1980), no. 1, 125–137.

    Article  MATH  MathSciNet  Google Scholar 

  56. L. A. Rubel and A. L. Shields, Invariant subspaces of analytic function spaces, Function Algebras (Proc. Internat. Sympos. on Function Algebras, Tulane Univ., 1965), Scott-Foresman, Chicago, Ill., 1966, pp. 341–342.

    Google Scholar 

  57. I. J. Schark, Maximal ideals in an algebra of bounded analytic functions, J. Math. Mech. 10 (1961), 735–746.

    MATH  MathSciNet  Google Scholar 

  58. C. F. Schubert, The corona theorem as an operator theorem, Proc. Amer. Math. Soc. 69 (1978), no. 1, 73–76.

    Article  MATH  MathSciNet  Google Scholar 

  59. Nessim Sibony, Prolongement analytique des fonctions holomorphes bornées (French), C. R. Acad. Sci. Paris 275 (1972), A973–A976.

    MathSciNet  Google Scholar 

  60. C. F. Schubert, Probléme de la couronne pour des domaines pseudoconvexes á bord lisse, Ann. of Math. (2)126 (1987), no. 3, 675–682 (French).

    Google Scholar 

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

    MATH  MathSciNet  Google Scholar 

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

    Google Scholar 

  63. V. A. Tolokonnikov, Blaschke products with the Carleson-Newman condition, and ideals of the algebra H ∞, Zap. Nauchn. Sem. Leningrad. Otdel. Mat. Inst. Steklov. (LOMI) 149 (1986), no. Issled. Linein. Teor. Funktsii. XV, 93–102, 188 (Russian, with English summary); English transl., J. Soviet Math. 42 (1988), no. 2, 1603–1610.

    Google Scholar 

  64. V. A. Tolokonnikov, Estimates in Carleson’s corona theorem and finitely generated ideals of the algebra H ∞, Funktsional. Anal. i Prilozhen. 14 (1980), no. 4, 85–86 (Russian).

    MathSciNet  Google Scholar 

  65. V. A. Tolokonnikov, Estimates in the Carleson corona theorem, ideals of the algebra H ∞ a problem of Sz.-Nagy, Zap. Nauchn. Sem. Leningrad. Otdel. Mat. Inst. Steklov. (LOMI) 113 (1981), 178–198, 267 (Russian, with English summary). Investigations on linear operators and the theory of functions, XI.

    Google Scholar 

  66. S. Treil, An operator Corona theorem, Indiana Univ. Math. J. 53 (2004), no. 6, 1763–1780, DOI 10.1512/iumj.2004.53.2640. MR2106344 (2005j:30067)

  67. S. Treil, The problem of ideals of H ∞: beyond the exponent 3/2, J. Funct. Anal. 253 (2007), no. 1, 220–240.

    Article  MATH  MathSciNet  Google Scholar 

  68. S. Treil, Lower bounds in the matrix Corona theorem and the codimension one conjecture, Geom. Funct. Anal. 14 (2004), no. 5, 1118–1133.

    Article  MATH  MathSciNet  Google Scholar 

  69. S. Treil, Estimates in the corona theorem and ideals of H ∞: a problem of T. Wolff, J. Anal. Math. 87 (2002), 481–495. Dedicated to the memory of Thomas H. Wolff.

    Google Scholar 

  70. S. 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 

  71. S. Treil, Angles between co-invariant subspaces, and the operator corona problem. The Szőkefalvi-Nagy problem, Dokl. Akad. Nauk SSSR 302 (1988), no. 5, 1063–1068 (Russian); English transl., Soviet Math. Dokl. 38 (1989), no. 2, 394–399.

    Google Scholar 

  72. Sergei Treil and Brett D.Wick, Analytic projections, corona problem and geometry of holomorphic vector bundles, J. Amer. Math. Soc. 22 (2009), no. 1, 55–76.

    Google Scholar 

  73. Sergei Treil and Brett D., The matrix-valued H p corona problem in the disk and polydisk, J. Funct. Anal. 226 (2005), no. 1, 138–172.

    Article  MATH  MathSciNet  Google Scholar 

  74. Sergei Treil and Brett D., A note on multiplier algebras on reproducing kernel Hilbert spaces, Proc. Amer. Math. Soc. 136 (2008), no. 8, 2835–2838.

    Google Scholar 

  75. Sergei Treil and Brett D., A vector-valued H p corona theorem on the polydisk, Integral Equations Operator Theory 56 (2006), no. 1, 129–149.

    Article  MathSciNet  Google Scholar 

  76. Sergei Treil and Brett D., An estimate for ideals in H ∞ (D), Integral Equations Operator Theory 53 (2005), no. 4, 573–587.

    Article  MathSciNet  Google Scholar 

  77. Sergei Treil and Brett D., A corona theorem for multipliers on Dirichlet space, Integral Equations Operator Theory 49 (2004), no. 1, 123–139.

    Article  MathSciNet  Google Scholar 

  78. Sergei Treil and Brett D., A new estimate for the vector valued corona problem, J. Funct. Anal. 189 (2002), no. 1, 267–282.

    Article  MathSciNet  Google Scholar 

  79. Tavan T. Trent and Debendra Banjade, Problem of Ideals on the Multiplier Algebra of the Dirichlet Space, preprint.

    Google Scholar 

  80. Tavan T. Trent and Brett D. Wick, Toeplitz corona theorems for the polydisk and the unit ball, Complex Anal. Oper. Theory 3 (2009), no. 3, 729–738.

    Google Scholar 

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

    Google Scholar 

  82. N. Th. Varopoulos, “BMO functions and the \(\bar{\partial }\) -equation”, Pacific J. Math. 71 (1977), no. 1, 221–273.

    Google Scholar 

  83. Pascale Vitse, A few more remarks on the operator valued corona problem, Acta Sci. Math. (Szeged) 69 (2003), no. 3–4, 831–852.

    MATH  MathSciNet  Google Scholar 

  84. N. Th. Varopoulos, A tensor product approach to the operator corona problem, J. Operator Theory 50 (2003), no. 1, 179–208.

    MathSciNet  Google Scholar 

  85. Jie Xiao, The \(\bar{\partial }\) -problem for multipliers of the Sobolev space, Manuscripta Math. 97 (1998), no. 2, 217–232.

    Article  MATH  MathSciNet  Google Scholar 

  86. Yuan Xu, In memoriam: Donald J. Newman (1930–2007), J. Approx. Theory 154 (2008), no. 1, 37–58.

    Article  MATH  MathSciNet  Google Scholar 

Download references

Acknowledgements

Research of Sergei Treil supported in part by National Science Foundation DMS grant # 0800876. Research of Brett Wick supported in part by National Science Foundation DMS grant # 955432.

Author information

Authors and Affiliations

  1. Department of Mathematics, Texas A&M University, College Station, TX, 77843, USA

    Ronald G. Douglas

  2. Department of Mathematics, Washington University in St. Louis, St. Louis, MO, 63130, USA

    Steven G. Krantz

  3. Department of Mathematics and Statistics, McMaster University, Hamilton, ON, L8S 4K1, Canada

    Eric T. Sawyer

  4. Department of Mathematics, Brown University, Providence, RI, 02912, USA

    Sergei Treil

  5. School of Mathematics, Georgia Institute of Technology, 686 Cherry Street, Atlanta, GA, 30332-0160, USA

    Brett D. Wicks

Authors
  1. Ronald G. Douglas
    View author publications

    You can also search for this author in PubMed Google Scholar

  2. Steven G. Krantz
    View author publications

    You can also search for this author in PubMed Google Scholar

  3. Eric T. Sawyer
    View author publications

    You can also search for this author in PubMed Google Scholar

  4. Sergei Treil
    View author publications

    You can also search for this author in PubMed Google Scholar

  5. Brett D. Wicks
    View author publications

    You can also search for this author in PubMed Google Scholar

Corresponding author

Correspondence to Ronald G. Douglas .

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

Douglas, R.G., Krantz, S.G., Sawyer, E.T., Treil, S., Wicks, B.D. (2014). A History of the Corona Problem. 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_1

Download citation

Publish with us

Policies and ethics

Search

Navigation