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Recent Progress and Open Problems in the Bergman Space

  • Alexandru Aleman
  • Håakan Hedenmalm
  • Stefan Richter
Conference paper
Part of the Operator Theory: Advances and Applications book series (OT, volume 156)

Abstract

The aim of this work is to provide a survey of interesting open problems in the theory of the Bergman spaces.

Keywords

Hardy Space Invariant Subspace Bergman Space Blaschke Product Extremal Function 
These keywords were added by machine and not by the authors. This process is experimental and the keywords may be updated as the learning algorithm improves.

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References

  1. [1]
    E. Abakumov and A. Borichev, Shift invariant subspaces with arbitrary indices inE. Abakumov and A. Borichev,lpspaces, J. Funct. Anal. 188 (2002), 1–26.CrossRefMathSciNetGoogle Scholar
  2. [2]
    A. Abkar, Norm approximation by polynomials in some weighted Bergman spaces. J. Funct. Anal. 191 (2002), no. 2, 224–240.CrossRefzbMATHMathSciNetGoogle Scholar
  3. [3]
    A. Abkar, H. Hedenmalm, A Riesz representation formula for super-biharmonic functions. Ann. Acad. Sci. Fenn. Math. 26 (2001), no. 2, 305–324.MathSciNetGoogle Scholar
  4. [4]
    J. Agler, Interpolation (unpublished manuscript).Google Scholar
  5. [5]
    J. Agler and J. McCarthy, Complete Nevanlinna-Pick kernels, J. Funct. Anal. 175 (2000), 111–124.CrossRefMathSciNetGoogle Scholar
  6. [6]
    A. Aleman, H. Hedenmalm, S. Richter, C. Sundberg, Curious properties of canonical divisors in weighted Bergman spaces, Entire functions in modern analysis [Boris Levin Memorial Conference] (Tel-Aviv, 1997), 1–10, Israel Math. Conf. Proc., 15, Bar-Ilan Univ., Ramat Gan, 2001.Google Scholar
  7. [7]
    A. Aleman and S. Richter, Some suffcient conditions for the division property of invariant subspaces in weighted Bergman spaces, J. Funct. Anal. 144 (1997), 542–556.CrossRefMathSciNetGoogle Scholar
  8. [8]
    A. Aleman and S. Richter, Single point extremal functions in Bergman-type spaces, Indiana Univ. Math. J. 51 (2002), 581–605.CrossRefMathSciNetGoogle Scholar
  9. [9]
    A. Aleman, S. Richter, and W. Ross, Pseudocontinuations and the backward shift. Indiana Univ. Math. J. 47 (1998), no. 1, 223–276.CrossRefMathSciNetGoogle Scholar
  10. [10]
    A. Aleman, S. Richter, C. Sundberg, Beurling’s theorem for the Bergman space, Acta Math. 177 (1996), 275–310.MathSciNetGoogle Scholar
  11. [11]
    A. Aleman, S. Richter, C. Sundberg, The majorization function and the index of invariant subspaces in the Bergman spaces. J. Anal. Math. 86 (2002), 139–182.MathSciNetGoogle Scholar
  12. [12]
    C. Apostol, H. Bercovici, C. Foiaş, C. Pearcy, Invariant subspaces, dilation theory, and the structure of the predual of a dual algebra, I, J. Funct. Anal. 63 (1985),369–404.CrossRefMathSciNetGoogle Scholar
  13. [13]
    W. Arveson, Subalgebras of C*-algebras, III. Multivariable operator theory. Acta Math. 181 (1998), 159–228.zbMATHMathSciNetGoogle Scholar
  14. [14]
    A. Atzmon, Entire functions, invariant subspaces and Fourier transforms. Proceedings of the Ashkelon Workshop on Complex Function Theory (1996), 37–52, Israel Math. Conf. Proc., 11, Bar-Ilan Univ., Ramat Gan, 1997.Google Scholar
  15. [15]
    A. Atzmon, Maximal, minimal, and primary invariant subspaces. J. Funct. Anal. 185 (2001), 155–213.CrossRefzbMATHMathSciNetGoogle Scholar
  16. [16]
    S. Bergman, The kernel function and conformal mapping. Second, revised edition. Mathematical Surveys, No. V. American Mathematical Society, Providence, R.I., 1970.Google Scholar
  17. [17]
    A. Beurling, On two problems concerning linear transformations in Hilbert space, Acta Math. 81 (1949), 39–255.Google Scholar
  18. [18]
    G. Bomash, A Blaschke-type product and random zero sets for the Bergman space, Ark. Mat. 30 (1992), 45–60.zbMATHMathSciNetGoogle Scholar
  19. [19]
    A.0 Borichev, Invariant subspaces of a given index in Banach spaces of analytic functions, J. Reine Angew. Math. 505 (1998), 23–44.zbMATHMathSciNetGoogle Scholar
  20. [20]
    A. Borichev, H. Hedenmalm, Harmonic functions of maximal growth: invertibility and cyclicity in Bergman spaces. J. Amer. Math. Soc. 10 (1997), 761–796.CrossRefMathSciNetGoogle Scholar
  21. [21]
    L. Brown, B. Korenblum, Cyclic vectors in A−∞, Proc. Amer. Math. Soc. 102 (1988), 137–138.MathSciNetGoogle Scholar
  22. [22]
    P.L. Duren, D. Khavinson, H.S. Shapiro, Extremal functions in invariant subspaces of Bergman spaces. Illinois J. Math. 40 (1996), 202–210.MathSciNetGoogle Scholar
  23. [23]
    P.L. Duren, D. Khavinson, H.S. Shapiro, C. Sundberg Contractive zero-divisors in Bergman spaces, Pacific J. Math. 157 (1993), 37–56.MathSciNetGoogle Scholar
  24. [24]
    P.L. Duren, D. Khavinson, H.S. Shapiro, C. Sundberg, Invariant subspaces and the biharmonic equation, Michigan Math. J. 41 (1994), 247–259.MathSciNetGoogle Scholar
  25. [25]
    P.L. Duren, A. Schuster, Bergman Spaces, Mathematical Surveys and Monographs, 100, American Mathematical Society, 2004.Google Scholar
  26. [26]
    M. Engliš, A Loewner-type lemma for weighted biharmonic operators, Pacific J. Math. 179 (1997), 343–353.MathSciNetGoogle Scholar
  27. [27]
    M. Engliš,Weighted biharmonic Green functions for rational weights, Glasgow Math. J. 41 (1999), 239–269.Google Scholar
  28. [28]
    P.R. Garabedian, A partial differential equation arising in conformal mapping, Pacific J. Math. 1 (1951), 485–524zbMATHMathSciNetGoogle Scholar
  29. [29]
    D.C.V. Greene, S. Richter, C. Sundberg, The structure of inner multipliers in spaces with complete Nevanlinna-Pick kernels, J. Funct. Anal. 194 (2002), no. 2, 311–331.CrossRefMathSciNetGoogle Scholar
  30. [30]
    J. Hadamard, Œuvres de Jacques Hadamard, Vols. 1–4, Editions du Centre National de la Recherche Scienti.que, Paris, 1968.Google Scholar
  31. [31]
    J. Hansbo, Reproducing kernels and contractive divisors in Bergman spaces. Zap. Nauchn. Sem. S.-Peterburg. Otdel. Mat. Inst. Steklov. (POMI) 232 (1996), Issled. po Linein. Oper. i Teor. Funktsii. 24, 174–198, 217; translation in J. Math. Sci. (New York) 92 (1998), 3657–3674.zbMATHGoogle Scholar
  32. [32]
    V.P. Havin, N.K. Nikolski (editors), Linear and complex analysis, Problem book 3, Part II, Lecture Notes in Mathematics 1574, Springer-Verlag, Berlin, 1994.Google Scholar
  33. [33]
    W.K. Hayman, On a conjecture of Korenblum. Analysis (Munich) 19 (1999), 195–205.zbMATHMathSciNetGoogle Scholar
  34. [34]
    H. Hedenmalm, A factorization theorem for square area-integrable analytic functions, J. Reine angew. Math. 422 (1991), 45–68.zbMATHMathSciNetGoogle Scholar
  35. [35]
    H. Hedenmalm, A factoring theorem for a weighted Bergman space, St. Petersburg Math. J. 4 (1993), 163–174.MathSciNetGoogle Scholar
  36. [36]
    H. Hedenmalm, An invariant subspace of the Bergman space having the codimension two property, J. Reine Angew. Math. 443 (1993), 1–9.zbMATHMathSciNetGoogle Scholar
  37. [37]
    H. Hedenmalm, Spectral properties of invariant subspaces in the Bergman space, J. Funct. Anal. 116 (1993), 441–448.CrossRefzbMATHMathSciNetGoogle Scholar
  38. [38]
    H. Hedenmalm, A factoring theorem for the Bergman space, Bull. London Math. Soc. 26 (1994), 113–126.zbMATHMathSciNetGoogle Scholar
  39. [39]
    H. Hedenmalm, A computation of Green functions for the weighted biharmonic operators Δ∣z−2αΔ,with α > −1, Duke Math. J. 75 (1994), 51–78.CrossRefzbMATHMathSciNetGoogle Scholar
  40. [40]
    H. Hedenmalm, Open problems in the function theory of the Bergman space, Festschrift in honour of Lennart Carleson and Yngve Domar (Uppsala, 1993), 153–169, Acta Univ. Upsaliensis Skr. Uppsala Univ. C Organ. Hist., 58, Uppsala Univ., Uppsala, 1995.Google Scholar
  41. [41]
    H. Hedenmalm, Boundary value problems for weighted biharmonic operators, St. Petersburg Math. J. 8 (1997), 661–674.MathSciNetGoogle Scholar
  42. [42]
    H. Hedenmalm, Recent progress in the function theory of the Bergman space. Holomorphic spaces (Berkeley, CA, 1995), 35–50, Math. Sci. Res. Inst. Publ., 33, Cambridge Univ. Press, Cambridge, 1998.Google Scholar
  43. [43]
    H. Hedenmalm, Recent developments in the function theory of the Bergman space. European Congress of Mathematics, Vol. I (Budapest, 1996), 202–217, Progr. Math., 168, Birkhäuser, Basel, 1998.Google Scholar
  44. [44]
    H. Hedenmalm, Maximal invariant subspaces in the Bergman space, Ark. Mat. 36 (1998), 97–101.zbMATHMathSciNetGoogle Scholar
  45. [45]
    H. Hedenmalm, An off-diagonal estimate of Bergman kernels, J. Math. Pures Appl. 79 (2000), 163–172.CrossRefzbMATHMathSciNetGoogle Scholar
  46. [46]
    H. Hedenmalm, B. Korenblum, K. Zhu, Theory of Bergman spaces. Graduate Texts in Mathematics, 199. Springer-Verlag, New York, 2000.Google Scholar
  47. [47]
    H. Hedenmalm, S. Jakobsson, S. Shimorin, A maximum principle à la Hadamard for biharmonic operators with applications to the Bergman spaces, C. R. Acad. Sci. Paris Sér. I Math. 328 (1999), 973–978.MathSciNetGoogle Scholar
  48. [48]
    H. Hedenmalm, S. Jakobsson, S. Shimorin, A biharmonic maximum principle for hyperbolic surfaces. J. Reine Angew. Math. 550 (2002), 25–75.MathSciNetGoogle Scholar
  49. [49]
    H. Hedenmalm, Y. Perdomo, Mean value surfaces with prescribed curvature form. J. Math. Pures Appl., to appear.Google Scholar
  50. [50]
    H. Hedenmalm, S. Richter, K. Seip, Interpolating sequences and invariant subspaces of given index in the Bergman spaces. J. Reine Angew. Math. 477 (1996), 13–30.MathSciNetGoogle Scholar
  51. [51]
    H. Hedenmalm, S. Shimorin, Hele-Shaw flow on hyperbolic surfaces. J. Math. Pures Appl. 81 (2002), 187–222.MathSciNetGoogle Scholar
  52. [52]
    H. Hedenmalm and K. Zhu,On the failure of optimal factorization for certain weighted Bergman spaces, Complex Var. Theory Appl. 19 (1992), 165–176.MathSciNetGoogle Scholar
  53. [53]
    A. Hinkkanen, On a maximum principle in Bergman space, J. Analyse Math. 79 (1999), 335–344.zbMATHMathSciNetGoogle Scholar
  54. [54]
    C. Horowitz, Zeros of functions in the Bergman spaces, Duke Math. J. 41 (1974), 693–710.CrossRefzbMATHMathSciNetGoogle Scholar
  55. [55]
    C. Horowitz, Factorization theorems for functions in the Bergman spaces, Duke Math. J. 44 (1977), 201–213.CrossRefzbMATHMathSciNetGoogle Scholar
  56. [56]
    P. Koosis, The logarithmic integral. I. Corrected reprint of the 1988 original. Cambridge Studies in Advanced Mathematics, 12. Cambridge University Press, Cambridge, 1998.Google Scholar
  57. [57]
    B. Korenblum, An extension of the Nevanlinna theory, Acta Math. 135 (1975), 187–219.zbMATHMathSciNetGoogle Scholar
  58. [58]
    B. Korenblum, A Beurling-type theorem, Acta Math. 138 (1977), 265–293.zbMATHMathSciNetGoogle Scholar
  59. [59]
    B. Korenblum, A maximum principle for the Bergman space, Publ. Mat. 35 (1991),479–486.zbMATHMathSciNetGoogle Scholar
  60. [60]
    B. Korenblum, Outer functions and cyclic elements in Bergman spaces. J. Funct. Anal. 115 (1993), no. 1, 104–118.CrossRefzbMATHMathSciNetGoogle Scholar
  61. [61]
    E. LeBlanc, A probabilistic zero set condition for Bergman spaces, Michigan Math. J. 37 (1990), 427–438.zbMATHMathSciNetGoogle Scholar
  62. [62]
    S. McCullough, S. Richter, Bergman-type reproducing kernels, contractive divisors and dilations, J. Funct. Anal. 190 (2002), 447–480.CrossRefMathSciNetGoogle Scholar
  63. [63]
    S. McCullough, T. Trent, Invariant subspaces and Nevanlinna-Pick kernels, J. Funct. Anal. 178 (2000), 226–249.CrossRefMathSciNetGoogle Scholar
  64. [64]
    S.N. Mergelyan, Weighted approximation by polynomials, American Mathematical Society Translations (series 2) 10 (1958), 59–106.zbMATHMathSciNetGoogle Scholar
  65. [65]
    B. Muckenhoupt, Weighted norm inequalities for the Hardy maximal function, Trans. Amer. Math. Soc. 165 (1972), 207–226.zbMATHMathSciNetGoogle Scholar
  66. [66]
    S. Richter, Invariant subspaces in Banach spaces of analytic functions, Trans. Amer. Math. Soc. 304 (1987), 585–616.zbMATHMathSciNetGoogle Scholar
  67. [67]
    S. Richter, Unitary equivalence of invariant subspaces of Bergman and Dirichlet spaces. Pacific J. Math. 133 (1988), no. 1, 151–156.zbMATHMathSciNetGoogle Scholar
  68. [68]
    S. Richter, A representation theorem for cyclic analytic two-isometries, Trans. Amer. Math. Soc. 328 (1991), 325–349.zbMATHMathSciNetGoogle Scholar
  69. [69]
    K. Seip, Beurling type density theorems in the unit disk, Invent. Math. 113 (1993), 21–39.CrossRefzbMATHMathSciNetGoogle Scholar
  70. [70]
    K. Seip, On a theorem of Korenblum, Ark. Mat. 32 (1994), 237–243.zbMATHMathSciNetGoogle Scholar
  71. [71]
    K. Seip, On Korenblum’s density condition for the zero sequences of A−α. J. Anal. Math. 67 (1995), 307–322.zbMATHMathSciNetGoogle Scholar
  72. [72]
    H.S. Shapiro, A.L. Shields, On the zeros of functions with finite Dirichlet integral and some related function spaces. Math. Z. 80 (1962), 217–229.CrossRefMathSciNetGoogle Scholar
  73. [73]
    A.L. Shields, Cyclic vectors in Banach spaces of analytic functions, Operators and Function Theory (Lancaster, 1984), S. C. Power (editor), NATO Adv. Sci. Inst. Ser. C Math. Phys. Sci., 153, Reidel, Dordrecht, 1985, 315–349.Google Scholar
  74. [74]
    S.M. Shimorin, Factorization of analytic functions in weighted Bergman spaces, St. Petersburg Math. J. 5 (1994), 1005–1022.MathSciNetGoogle Scholar
  75. [75]
    S.M. Shimorin, On a family of conformally invariant operators, St. Petersburg Math. J. 7 (1996), 287–306.MathSciNetGoogle Scholar
  76. [76]
    S.M. Shimorin, The Green function for the weighted biharmonic operatorΔ(1 − ∣z2)−2αΔand the factorization of analytic functions (Russian), Zap. Nauchn. Sem. S.-Peterburg. Otdel. Mat. Inst. Steklov. (POMI) 222 (1995), Issled. po Linein. Oper. i Teor. Funktsii 23, 203–221.zbMATHGoogle Scholar
  77. [77]
    S.M. Shimorin, Single point extremal functions in weighted Bergman spaces, Nonlinea boundary-value problems and some questions of function theory, J. Math. Sci. 80 (1996), 2349–2356.MathSciNetGoogle Scholar
  78. [78]
    S.M. Shimorin, The Green functions for weighted biharmonic operators of the form Δw−1Δ in the unit disk, Some questions of mathematical physics and function theory, J. Math. Sci. (New York) 92 (1998), 4404–4411.MathSciNetGoogle Scholar
  79. [79]
    S.M. Shimorin, Approximate spectral synthesis in the Bergman space. Duke Math. J. 101 (2000), no. 1, 1–39.CrossRefzbMATHMathSciNetGoogle Scholar
  80. [80]
    S.M. Shimorin, Wold-type decompositions and wandering subspaces for operators close to isometries, J. Reine Angew. Math. 531 (2001), 147–189.zbMATHMathSciNetGoogle Scholar
  81. [81]
    S.M. Shimorin, An integral formula for weighted Bergman reproducing kernels, Complex Var. Theory Appl. 47 (2002), no. 11, 1015–1028.zbMATHMathSciNetGoogle Scholar
  82. [82]
    S.M. Shimorin, On Beurling-type theorems in weighted l2 and Bergman spaces. Proc. Amer. Math. Soc. 131 (2003), no. 6, 1777–1787.CrossRefzbMATHMathSciNetGoogle Scholar
  83. [83]
    J. Thomson, and L. Yang, Invariant subspaces with the codimension one property in Lta(μ. Indiana Univ. Math. J. 44 (1995), 1163–1173.CrossRefMathSciNetGoogle Scholar
  84. [84]
    R.J. Weir, Construction of Green functions for weighted biharmonic operators. J. Math. Anal. Appl. 288 (2003), no. 2, 383–396.CrossRefzbMATHMathSciNetGoogle Scholar
  85. [85]
    Z. Wu and L. Yang, The codimension-1 property in Bergman spaces over planar regions. Michigan Math. J. 45 (1998), 369–373.MathSciNetGoogle Scholar
  86. [86]
    L. Yang, Invariant subspaces of the Bergman space and some subnormal operators in \(\mathbb{A}_1 \backslash \mathbb{A}_2\). Michigan Math. J. 42 (1995), 301–310.zbMATHMathSciNetGoogle Scholar

Copyright information

© Birkhäuser Verlag Basel/Switzerland 2005

Authors and Affiliations

  • Alexandru Aleman
    • 1
  • Håakan Hedenmalm
    • 2
  • Stefan Richter
    • 3
  1. 1.Center for MathematicsLund UniversityLundSweden
  2. 2.Department of MathematicsThe Royal Institute of TechnologyStockholmSweden
  3. 3.Department of MathematicsUniversity of TennesseeKnoxvilleUSA

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