The Halmos Similarity Problem

  • Gilles Pisier
Part of the Operator Theory Advances and Applications book series (OT, volume 207)


We describe the ideas that led to the formulation of the problem and its solution, as well as some related questions left open.

Mathematics Subject Classification (2000)

Primary 47A20 47B35 Secondary 46L07 47L25 


Similarity polynomially bounded completely bounded Hankel operator length 


Unable to display preview. Download preview PDF.

Unable to display preview. Download preview PDF.


  1. [1]
    J. Agier. Rational dilation on an annulus. Ann. of Math. (2) 121 (1985), no. 3, 537–563.Google Scholar
  2. [2]
    J. Agler, J. Harland and B.J. Raphael. Classical function theory, operator dilation theory, and machine computation on multiply-connected domains. Mem. Amer. Math. Soc. 191 (2008), no. 892, viii+159 pp.Google Scholar
  3. [3]
    A.B. Aleksandrov and V. Peller, Hankel Operators and Similarity to a Contraction. Int. Math. Res. Not. 6 (1996), 263–275.CrossRefMathSciNetGoogle Scholar
  4. [4]
    A. Aleman and O. Constantin, Hankel operators on Bergman spaces and similarity to contractions. Int. Math. Res. Not. 35 (2004), 1785–1801.CrossRefMathSciNetGoogle Scholar
  5. [5]
    T. Ando, On a Pair of Commutative Contractions. Acta Sci. Math. 24 (1963), 88–90.zbMATHMathSciNetGoogle Scholar
  6. [6]
    W. Arveson. Dilation theory yesterday and today. Oper. Theory Adv. Appl. 207 (2010), 99–123 (in this volume).MathSciNetGoogle Scholar
  7. [7]
    C. Badea and V.I. Paulsen, Schur Multipliers and Operator-Valued Foguel—Hankel Operators. Indiana Univ. Math. J. 50 (2001), no. 4, 1509–1522.CrossRefzbMATHMathSciNetGoogle Scholar
  8. [8]
    J. Bourgain, On the Similarity Problem for Polynomially Bounded Operators on Hilbert Space. Israel J. Math. 54 (1986), 227–241.CrossRefzbMATHMathSciNetGoogle Scholar
  9. [9]
    M. Bozejko, Littlewood Functions, Hankel Multipliers and Power Bounded Operators on a Hilbert Space. Colloquium Math. 51 (1987), 35–42.zbMATHMathSciNetGoogle Scholar
  10. [10]
    J. Carlson, D. Clark, C. Foias and J.P. Williams, Projective Hilbert A(D) -Modules. New York J. Math. 1 (1994), 26–38, electronic.zbMATHMathSciNetGoogle Scholar
  11. [11]
    J.F. Carlson and D.N. Clark, Projectivity and Extensions of Hilbert Modules Over A(D N). Michigan Math. J. 44 (1997), 365–373.CrossRefzbMATHMathSciNetGoogle Scholar
  12. [12]
    J.F. Carlson and D.N. Clark, Cohomology and Extensions of Hilbert Modules. J. Funct. Anal. 128 (1995), 278–306.CrossRefzbMATHMathSciNetGoogle Scholar
  13. [13]
    O. Constantin and F. Jaëck, A joint similarity problem on vector-valued Bergman spaces. J. Funct. Anal. 256 (2009), 2768–2779.CrossRefzbMATHMathSciNetGoogle Scholar
  14. [14]
    K.R. Davidson and V.I. Paulsen, Polynomially bounded operators. J. Reine Angew. Math. 487 (1997), 153–170.zbMATHMathSciNetGoogle Scholar
  15. [15]
    A.M. Davie, Quotient Algebras of Uniform Algebras. J. London Math. Soc. 7 (1973), 31–40.CrossRefzbMATHMathSciNetGoogle Scholar
  16. [16]
    R.G. Douglas and V.I. Paulsen. Hilbert modules over function algebras. Pitman Research Notes in Mathematics Series, 217. John Wiley & Sons, Inc., New York, 1989.Google Scholar
  17. [17]
    M.A. Dritschel and S. McCullough. The failure of rational dilation on a triply connected domain. J. Amer. Math. Soc. 18 (2005), 873–918.CrossRefzbMATHMathSciNetGoogle Scholar
  18. [18]
    S. Drury, Remarks on von Neumann’s Inequality. Banach Spaces, Harmonic Analysis, and Probability Theory. Proceedings, R. Blei and S. Sydney (eds.), Storrs 80/81, Springer Lecture Notes 995, 14–32.Google Scholar
  19. [19]
    S. Foguel, A Counterexample to a Problem of Sz.-Nagy. Proc. Amer. Math. Soc. 15 (1964), 788–790.CrossRefzbMATHMathSciNetGoogle Scholar
  20. [20]
    C. Foias, Sur Certains Théorèmes de J. von Neumann Concernant les Ensembles Spectraux. Acta Sci. Math. 18 (1957), 15–20.zbMATHMathSciNetGoogle Scholar
  21. [21]
    C. Foias and J.P. Williams, On a Class of Polynomially Bounded Operators. Preprint (unpublished, approximately 1976).Google Scholar
  22. [22]
    C. Foias and I. Suciu. On operator representation of logmodular algebras. Bull. Acad. Polon. Sci. Sér. Sci. Math. Astronom. Phys. 16 (1968) 505–509.zbMATHMathSciNetGoogle Scholar
  23. [23]
    J.B. Garnett. Bounded analytic functions. Pure and Applied Mathematics, 96. Academic Press, Inc., New York-London, 1981.Google Scholar
  24. [24]
    U. Haagerup, Solution of the Similarity Problem for Cyclic Representations of C*-Algebras. Annals of Math. 118 (1983), 215–240.CrossRefMathSciNetGoogle Scholar
  25. [25]
    P. Halmos, On Foguel’s answer to Nagy’s question. Proc. Amer. Math. Soc. 15 1964 791–793.zbMATHMathSciNetGoogle Scholar
  26. [26]
    P. Halmos, Ten Problems in Hilbert Space. Bull. Amer. Math. Soc. 76 (1970), 887–933.CrossRefzbMATHMathSciNetGoogle Scholar
  27. [27]
    R. Kadison. On the orthogonalization of operator representations. Amer. J. Math. 77 (1955), 600–620.CrossRefzbMATHMathSciNetGoogle Scholar
  28. [28]
    N. Kalton and C. Le Merdy, Solution of a Problem of Peller Concerning Similarity. J. Operator Theory 47 (2002), no. 2, 379–387.zbMATHMathSciNetGoogle Scholar
  29. [29]
    S. Kislyakov, Operators that are (dis)Similar to a Contraction: Pisier’s Counterexample in Terms of Singular Integrals (Russian). Zap. Nauchn. Zap. Nauchn. Sem. St.-Petersburg. Otdel. Mat. Inst. Steklov. (POMI) 247 (1997), Issled. po Linein. Oper, i Teor. Funkts. 25, 79–95, 300; translation in J. Math. Sci. (New York) 101 (2000), no. 3, 3093–3103.Google Scholar
  30. [30]
    S. Kislyakov, The similarity problem for some martingale uniform algebras (Russian). Zap. Nauchn. Sem. St.-Petersburg. Otdel. Mat. Inst. Steklov. (POMI) 270 (2000), Issled. po Linein. Oper, i Teor. Funkts. 28, 90–102, 365; translation in J. Math. Sci. (N. Y.) 115 (2003), no. 2, 2141–2146.Google Scholar
  31. [31]
    A. Lebow, A Power Bounded Operator which is not Polynomially Bounded. Mich. Math. J. 15 (1968), 397–399.CrossRefzbMATHMathSciNetGoogle Scholar
  32. [32]
    C. Le Merdy, The Similarity Problem for Bounded Analytic Semigroups on Hilbert Space. Semigroup Forum 56 (1998), 205–224.CrossRefzbMATHMathSciNetGoogle Scholar
  33. [33]
    B. Lotto, von Neumann’s Inequality for Commuting, Diagonalizable Contractions, I. Proc. Amer. Math. Soc. 120 (1994), 889–895.zbMATHMathSciNetGoogle Scholar
  34. [34]
    B. Lotto and T. Steger, von Neumann’s Inequality for Commuting, Diagonalizable Contractions, II. Proc. Amer. Math. Soc. 120 (1994), 897–901.zbMATHMathSciNetGoogle Scholar
  35. [35]
    J.E. McCarthy. On Pisier’s Construction. arXiv:math/9603212.Google Scholar
  36. [36]
    G. Misra and S. Sastry, Completely Contractive Modules and Associated Extremal Problems. J. Funct. Anal. 91 (1990), 213–220.CrossRefzbMATHMathSciNetGoogle Scholar
  37. [37]
    G. Misra and S. Sastry, Bounded Modules, Extremal Problems and a Curvature Inequality. J. Funct. Anal. 88 (1990), 118–134.CrossRefzbMATHMathSciNetGoogle Scholar
  38. [38]
    J. von Neumann, Eine Spektraltheorie für allgemeine Operatoren eines unitären Raumes. Math. Nachr. 4 (1951), 258–281.zbMATHMathSciNetGoogle Scholar
  39. [39]
    N. Nikolskii, Treatise on the Shift Operator. Springer Verlag, Berlin, 1986.Google Scholar
  40. [40]
    E.W. Packel. A semigroup analogue of Foguel’s counterexample. Proc. AMS 21 (1969) 240–244.zbMATHMathSciNetGoogle Scholar
  41. [41]
    S. Parrott, Unitary Dilations for Commuting Contractions. Pacific J. Math. 34 (1970), 481–490.zbMATHMathSciNetGoogle Scholar
  42. [42]
    S. Parrott, On a Quotient Norm and the Sz.-Nagy—Foias Lifting Theorem. J. Funct. Anal. 30 (1978), 311–328.CrossRefzbMATHMathSciNetGoogle Scholar
  43. [43]
    V. Paulsen, Every Completely Polynomially Bounded Operator is Similar to a Contraction. J. Funct. Anal. 55 (1984), 1–17.CrossRefzbMATHMathSciNetGoogle Scholar
  44. [44]
    V. Paulsen, Completely Bounded Homomorphisms of Operator Algebras. Proc. Amer. Math. Soc. 92 (1984), 225–228.CrossRefzbMATHMathSciNetGoogle Scholar
  45. [45]
    V. Paulsen, Completely Bounded Maps and Dilations. Pitman Research Notes in Math. 146, Longman, Wiley, New York, 1986.Google Scholar
  46. [46]
    V.l. Paulsen, Completely bounded maps and operator algebras. Cambridge Studies in Advanced Mathematics, 78. Cambridge University Press, Cambridge, 2002.Google Scholar
  47. [47]
    V.I. Paulsen and M. Raghupathi. Representations of logmodular algebras. Preprint, 2008.Google Scholar
  48. [48]
    V. Peller, Estimates of Functions of Power Bounded Operators on Hilbert Space. J. Oper. Theory 7 (1982), 341–372.zbMATHMathSciNetGoogle Scholar
  49. [49]
    G. Pisier, A Polynomially Bounded Operator on Hilbert Space which is not Similar to a Contraction. J. Amer. Math. Soc. 10 (1997), 351–369.CrossRefzbMATHMathSciNetGoogle Scholar
  50. [50]
    G. Pisier, The Similarity Degree of an Operator Algebra. St. Petersburg Math. J. 10 (1999), 103–146.MathSciNetGoogle Scholar
  51. [51]
    G. Pisier, Joint Similarity Problems and the Generation of Operator Algebras with Bounded Length. Integr. Equ. Op. Th. 31 (1998), 353–370.CrossRefzbMATHMathSciNetGoogle Scholar
  52. [52]
    G. Pisier, The Similarity Degree of an Operator Algebra II. Math. Zeit. 234 (2000), 53–81.CrossRefzbMATHMathSciNetGoogle Scholar
  53. [53]
    G. Pisier, Multipliers of the Hardy Space H 1 and Power Bounded Operators. Colloq. Math. 88 (2001), no. 1, 57–73.CrossRefzbMATHMathSciNetGoogle Scholar
  54. [54]
    G. Pisier, Similarity problems and completely bounded maps. Second, Expanded Edition. Springer Lecture Notes 1618 (2001).Google Scholar
  55. [55]
    G. Pisier, Introduction to operator space theory. London Mathematical Society Lecture Note Series, 294. Cambridge University Press, Cambridge, 2003.Google Scholar
  56. [56]
    E. Ricard, On a Question of Davidson and Paulsen. J. Funct. Anal. 192 (2002), no. 1, 283–294.CrossRefzbMATHMathSciNetGoogle Scholar
  57. [57]
    F. Riesz and B. Sz.-Nagy, Leçons d’Analyse Fonctionnelle. Gauthier—Villars, Paris, Akadémiai Kiadó, Budapest, 1965.Google Scholar
  58. [58]
    G.C. Rota, On Models for Linear Operators. Comm. Pure Appl. Math. 13 (1960), 468–472.CrossRefMathSciNetGoogle Scholar
  59. [59]
    B. Sz.-Nagy, On Uniformly Bounded Linear Transformations on Hilbert Space. Acta Sci. Math. (Szeged) 11 (1946-1948), 152–157.Google Scholar
  60. [60]
    B. Sz.-Nagy, Completely Continuous Operators with Uniformly Bounded Iterates. Publ. Math. Inst. Hungarian Acad. Sci. 4 (1959), 89–92.CrossRefzbMATHMathSciNetGoogle Scholar
  61. [61]
    B. Sz.-Nagy, Sur les Contractions de l’espace de Hilbert. Acta Sci. Math. 15 (1953), 87–92.zbMATHMathSciNetGoogle Scholar
  62. [62]
    B. Sz.-Nagy, Spectral Sets and Normal Dilations of Operators. Proc. Intern. Congr. Math. (Edinburgh, 1958), 412–422, Cambridge Univ. Press.Google Scholar
  63. [63]
    B. Sz.-Nagy and C. Foias, Harmonic Analysis of Operators on Hilbert Space. Akadémiai Kiadó, Budapest, 1970.Google Scholar
  64. [64]
    N. Varopoulos, On an Inequality of von Neumann and an Application of the Metric Theory of Tensor Products to Operators Theory. J. Funct. Anal. 16 (1974), 83–100.CrossRefzbMATHMathSciNetGoogle Scholar

Copyright information

© Springer Basel AG 2010

Authors and Affiliations

  • Gilles Pisier
    • 1
    • 2
  1. 1.Texas A&M UniversityCollege StationUSA
  2. 2.Institut Math. Jussieu Équipe d’Analyse Fonctionnelle, Case 186Université Paris VIParis Cedex 05France

Personalised recommendations