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Methods of Kreĭn Space Operator Theory

  • James Rovnyak
Conference paper
Part of the Operator Theory: Advances and Applications book series (OT, volume 134)

Abstract

This paper is a survey of old and recent methods of Krein space operator theory centering around Julia operators, extension problems for contraction operators, Hermitian kernels, and uniqueness questions. Examples related to coefficient problems for univalent functions are briefly discussed.

Keywords

Hilbert Space Selfadjoint Operator Contraction Operator Defect Operator KreIn Space 
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References

  1. [1]
    J. Agler and M. Stankusm-isometric transformations of Hilbert space. I II IIIIntegral Equations and Operator Theory 21 (1995), no. 4, 383–429, ibid. 23 (1995), no. 1, 1–48, ibid. 24 (1996), no. 4, 379–421.Google Scholar
  2. [2]
    D. AlpayReproducing kernel Krein spaces of analytic functions and inverse scatteringPh.D. thesis, Weizmann Institute of Science, 1985.Google Scholar
  3. [3]
    D. AlpayDilatations des commutants d’opérateurs pour des espaces de Krein de fonc-tions analytiques,Ann. Inst. Fourier (Grenoble) 39 (1989), no. 4, 1073–1094.MathSciNetCrossRefGoogle Scholar
  4. [4]
    D. AlpaySome remarks on reproducing kernel Krein spacesRocky Mountain J. Math. 21 (1991), no. 4, 1189–1205.MathSciNetzbMATHCrossRefGoogle Scholar
  5. [5]
    D. Alpay, A. Dijksma, J. Rovnyak, and H.S.V. de SnooSchur functions operator colligations,and reproducing kernel Pontryagin spacesOper. Theory Adv. Appl., vol. 96, Birkhäuser, Basel, 1997.Google Scholar
  6. [6]
    D. Alpay, A. Dijksma, J. Rovnyak, and H.S.V. de SnooReproducing kernel Pontryagin spacesHolomorphic Spaces, MSRI Publica-tions, vol. 33, Cambridge University Press, Cambridge, 1998, pp. 425–444.Google Scholar
  7. [7]
    R. Arocena, T.Ya. Azizov, A. Dijksma, and S.A.M. MarcantogniniOn commutant lifting with finite defectJ. Operator Theory 35 (1996), no. 1, 117–132.MathSciNetzbMATHGoogle Scholar
  8. [8]
    R. Arocena, T.Ya. Azizov, A. Dijksma, and S.A.M. MarcantogniniOn commutant lifting with finite defectII,J. Funct. Anal. 144 (1997), no. 1, 105–116.MathSciNetzbMATHCrossRefGoogle Scholar
  9. [9]
    N. AronszajnTheory of reproducing kernelsTrans. Amer. Math. Soc. 68 (1950), 337–404.MathSciNetzbMATHCrossRefGoogle Scholar
  10. [10]
    Gr. Arsene, T. Constantinescu, and A. GheondeaLifting of operators and prescribed numbers of negative squaresMichigan Math. J. 34 (1987), no. 2, 201–216.MathSciNetzbMATHGoogle Scholar
  11. [11]
    T.Ya. Azizov, Yu.P. Ginzburg, and H. LangerOn the work of M. G. Krein in the theory of spaces with an indefinite metricUkraïn. Mat. Zh. 46 (1994), no. 1–2, 5–17.MathSciNetGoogle Scholar
  12. [12]
    T.Ya. Azizov and I.S. IokhvidovLinear operators in spaces with an indefinite metricJohn Wiley & Sons Ltd., Chichester, 1989.Google Scholar
  13. [13]
    J.A. Ball and J.W. HeltonA Beurling-Lax theorem for the Lie group U(m n) which contains most classical interpolation theory J. Operator Theory 9 (1983), no. 1, 107–142.MathSciNetGoogle Scholar
  14. [14]
    J. BognárIndefinite inner product spacesSpringer-Verlag, New York, 1974, Ergebnisse der Mathematik und ihrer Grenzgebiete, Band 78.zbMATHCrossRefGoogle Scholar
  15. [15]
    L. de BrangesSquare summable power seriesUnpublished book ms., 1985.Google Scholar
  16. [16]
    L. de BrangesA proof of the Bieberbach conjecture,Acta Math. 154 (1985), no. 1–2, 137–152.CrossRefGoogle Scholar
  17. [17]
    L. de BrangesUnitary linear systems whose transfer functions are Riemann mapping func-tionsOperator theory and systems (Amsterdam, 1985), Oper. Theory Adv. Appl., vol. 19, Birkhäuser, Basel, 1986, pp. 105–124.Google Scholar
  18. [18]
    L. de BrangesComplementation in Krein spaces,Trans. Amer. Math. Soc. 305 (1988), no. 1, 277–291.MathSciNetzbMATHCrossRefGoogle Scholar
  19. [19]
    L. de BrangesUnderlying concepts in the proof of the Bieberbach conjectureUnderlying concepts in the proof of the Bieberbach conjecture (Berkeley, California, 1986), Amer. Math. Soc., Providence, RI, 1988, pp. 25–42.zbMATHGoogle Scholar
  20. [20]
    L. de BrangesA construction of Krein spaces of analytic functionsJ. Funct. Anal. 98 (1991), no. 1, 1–41.MathSciNetzbMATHCrossRefGoogle Scholar
  21. [21]
    G. Christner, Kin Y. Li, and J. RovnyakJulia operators and coefficient problemsNonselfadjoint operators and related topics (Beer Sheva, 1992), Oper. Theory Adv. Appl., vol. 73, Birkhäuser, Basel, 1994, pp. 138–181.Google Scholar
  22. [22]
    T. Constantinescu and A. GheondeaMinimal signature in lifting of operators. IJ. Operator Theory 22 (1989), no. 2, 345–367.MathSciNetzbMATHGoogle Scholar
  23. [23]
    T. Constantinescu and A. GheondeaExtending factorizations and minimal negative signaturesJ. Operator The-ory 28 (1992), no. 2, 371–402.Google Scholar
  24. [24]
    T. Constantinescu and A. GheondeaMinimal signature in lifting of operatorsII J. Funct. Anal. 103 (1992), no. 2, 317–351.MathSciNetzbMATHCrossRefGoogle Scholar
  25. [25]
    T. Constantinescu and A. GheondeaElementary rotations of linear operators in Krein spacesJ. Operator Theory 29 (1993), no. 1, 167–203.MathSciNetGoogle Scholar
  26. [26]
    T. Constantinescu and A. GheondeaRepresentations of Hermitian kernels by means of Krein spacesPubl. Res. Inst. Math. Sci. 33 (1997), no. 6, 917–951.MathSciNetCrossRefGoogle Scholar
  27. [27]
    B. Curgus and H. LangerOn a paper of de Brangespreprint, 1990.Google Scholar
  28. [28]
    Ch. DavisJ-unitary dilation of a general operatorActa Sci. Math. (Szeged) 31 (1970), 75–86.zbMATHGoogle Scholar
  29. [29]
    Ch. Davis and C. FoiasOperators with bounded characteristic function and their J-unitary dilationActa Sci. Math. (Szeged) 32 (1971), 127–139.MathSciNetzbMATHGoogle Scholar
  30. [30]
    A. Dijksma, M. Dritschel, S.A.M. Marcantognini, and H.S.V. de SnooThe commutant lifting theorem for contractions on Krein spacesOperator extensions, interpolation of functions and related topics (Timisoara, 1992), Oper. Theory Adv. Appl., vol. 61, Birkhäuser, Basel, 1993, pp. 65–83.Google Scholar
  31. [31]
    D. Dreibelbis, J. Rovnyak, and Kin Y. LiCoefficients of bounded univalent functionsProceedings of the Conference on Complex Analysis (Tianjin, 1992) (Cambridge, MA), Internat. Press, 1994, pp. 45–58.Google Scholar
  32. [[32]
    M.A. DritschelExtension theorems for operators on Krein spacesPh.D. thesis, University of Virginia, 1989.Google Scholar
  33. [33]
    M.A. DritschelCommutant lifting when the intertwining operator is not necessarily a con-traction,unpublished paper, 1993.Google Scholar
  34. [34]
    M.A. DritschelThe essential uniqueness property for operators on Krein spacesJ. Funct. Anal. 118 (1993), no. 1, 198–248.MathSciNetzbMATHCrossRefGoogle Scholar
  35. [35]
    M.A. DritschelA module approach to commutant lifting on Krein spaces,Operator the-ory, system theory and related topics. The Moshe Livsic anniversary volume, Oper. Theory Adv. Appl., vol. 123, Birkhäuser, Basel, 2001, pp. 195–206.MathSciNetGoogle Scholar
  36. [36]
    M.A. Dritschel and J. RovnyakOperators on indefinite inner product spacesLectures on operator theory and its applications (Waterloo, ON, 1994), Amer. Math. Soc., Providence, RI, pp. 141–232, 1996. Supplement and errata Google Scholar
  37. [37]
    M.A. Dritschel and J. RovnyakExtension theorems for contraction operators on Krein spacesExtension and interpolation of linear operators and matrix functions, Oper. Theory Adv. Appl., vol. 47, Birkhäuser, Basel, 1990, pp. 221–305.MathSciNetGoogle Scholar
  38. [38]
    M.A. Dritschel and J. RovnyakJulia operators and complementation in Krein spaces,Indiana Univ. Math. J. 40 (1991), no. 3, 885–901.MathSciNetzbMATHCrossRefGoogle Scholar
  39. [39]
    H. DymJ contractive matrix functions reproducing kernel Hilbert spaces and interpolationCBMS Regional Conference Series in Math., vol. 71, Amer. Math. Soc., Providence, RI, 1989.Google Scholar
  40. [40]
    C. Foias and A.E. FrazhoThe commutant lifting approach to interpolation problemsOper. Theory Adv. Appl., vol. 44, Birkhäuser, Basel, 1990.Google Scholar
  41. [41]
    C. Foias, A.E. Frazho, I. Gohberg, and M.A. KaashoekMetric constrained interpolation commutant lifting and systemsOper. Theory Adv. Appl., vol. 100, Birkhäuser, Basel, 1998.CrossRefGoogle Scholar
  42. [42]
    A. GheondeaContractive intertwining dilations of quasi-contractionsZ. Anal. Anwendungen 15 (1996), no. 1, 31–44.MathSciNetzbMATHGoogle Scholar
  43. [43]
    S. GhosechowdhuryAn expansion theorem for state space of unitary linear system whose transfer function is a Riemann mapping functionReproducing kernels and their applications (Newark, DE, 1997), Kluwer Acad. Publ., Dordrecht, 1999, pp. 8195.Google Scholar
  44. [44]
    S. GhosechowdhuryLöwner expansionsMath. Nachr. 210 (2000), 111–126.MathSciNetzbMATHCrossRefGoogle Scholar
  45. [45]
    T. HaraOperator inequalities and construction of Krein spacesIntegral Equations and Operator Theory 15 (1992), no. 4, 551–567.MathSciNetCrossRefGoogle Scholar
  46. [46]
    C. HellingsTwo-isometries on Pontryagin spacesPh.D. thesis, University of Virginia, 2000.Google Scholar
  47. [47]
    I.S. Iokhvidov, M.G. Krein, and H. LangerIntroduction to the spectral theory of operators in spaces with an indefinite metricAkademie-Verlag, Berlin, 1982.Google Scholar
  48. [48]
    A.N. KolmogorovStationary sequences in Hilbert spacesVestnik Moskov. Univ. Ser. I, Mat. Mech. 6 (1941), 1–40.Google Scholar
  49. [49]
    M.G. Krein and H. LangerOber die verallgemeinerten Resolventen und die charakteristische Funktion eines isometrischen Operators im RaumeIlk, Hilbert space operators and operator algebras (Proc. Internat. Conf., Tihany, 1970), North-Holland, Amsterdam, 1972, pp. 353–399. Colloq. Math. Soc. János Bolyai, 5.MathSciNetGoogle Scholar
  50. [50]
    M.G. Krein and H. Langerüber einige Fortsetzungsprobleme die eng mit der Theorie hermitescher Operatoren im Raume H zusammenhängen. I. Einige Funktionenklassen und ihre DarstellungenMath. Nachr. 77 (1977), 187–236.MathSciNetzbMATHGoogle Scholar
  51. [51]
    Kin Y. Li and J. RovnyakOn the coefficients of Riemann mappings of the unit disk into itselfContributions to operator theory and its applications, Oper. Theory Adv. Appl., vol. 62, Birkhäuser, Basel, 1993, pp. 145–163.Google Scholar
  52. [52]
    S.A.M. MarcantogniniThe commutant lifting theorem in the Krein space setting: a proof based on the coupling method,Indiana Univ. Math. J. 41 (1992), no. 4, 1303–1314.MathSciNetzbMATHCrossRefGoogle Scholar
  53. [53]
    M. Martin and M. PutinarLectures on hyponormal operatorsOper. Theory Adv. Appl., vol. 39, Birkhäuser, Basel, 1989.CrossRefGoogle Scholar
  54. [54]
    S.A. McCullough and L. RodmanTwo-selfadjoint operators in Krein spacesIntegral Equations and Operator Theory 26 (1996), no. 2, 202–209.MathSciNetzbMATHCrossRefGoogle Scholar
  55. [55]
    S.A. McCullough and L. RodmanHereditary classes of operators and matrices,Amer. Math. Monthly 104 (1997), no. 5, 415–430.MathSciNetzbMATHCrossRefGoogle Scholar
  56. [56]
    B.W. McEnnisCharacteristic functions and dilations of noncontractionsJ. Operator Theory 3 (1980), no. 1, 71–87.MathSciNetzbMATHGoogle Scholar
  57. [57]
    B.W. McEnnisModels for operators with bounded characteristic function,Acta Sci. Math. (Szeged) 43 (1981), no. 1–2, 71–90.MathSciNetzbMATHGoogle Scholar
  58. [58]
    B.W. McEnnisShifts on Krein spacesOperator theory: operator algebras and applications, Part 2 (Durham, NH, 1988), Amer. Math. Soc., Providence, RI, 1990, pp. 201–211.Google Scholar
  59. [59]
    N.K. Nikolskii and V.I. VasyuninQuasi-orthogonal decompositions with respect to complementary metrics and estimates for univalent functionsAlgebra i Analiz 2 (1990), no. 4, 1–81, Engl. transi., Leningrad Math. J. 2 (1991), no. 4, 691–764.MathSciNetGoogle Scholar
  60. [60]
    N.K. Nikolskii and V.I. VasyuninOperator-valued measures and coefficients of univalent functionsAlgebra i Analiz 3 (1991), no. 6, 1–75 (1992), Engl. transl., St. Petersburg Math. J. 3 (1992), no. 6, 1199–1270.Google Scholar
  61. [61]
    A. PitsillidesMathematica programREU project, available along with the web version of this survey at http://www.people.virginia.edu/jlr5m/papers/papers.html
  62. [62]
    S. RichterInvariant subspaces of the Dirichlet shiftJ. Reine Angew. Math. 386 (1988), 205–220.MathSciNetzbMATHGoogle Scholar
  63. [63]
    S. RichterA representation theorem for cyclic analytic two-isometriesTrans. Amer. Math. Soc. 328 (1991), no. 1, 325–349.MathSciNetzbMATHCrossRefGoogle Scholar
  64. [64]
    S. Richter and C. SundbergMultipliers and invariant subspaces in the Dirichlet spaceJ. Operator Theory 28 (1992), no. 1, 167–186.MathSciNetzbMATHGoogle Scholar
  65. [65]
    M. Rosenblum and J. RovnyakTopics in Hardy classes and univalent functionsBirkhäuser, Basel, 1994.zbMATHCrossRefGoogle Scholar
  66. [66]
    J. RovnyakAn extension problem for the coefficients of Riemann mappings, Seminar lecture, http://www.people.virginia.edu/~jlr5m/papers/papers.html.Google Scholar
  67. [67]
    J. RovnyakCoefficient estimates for Riemann mapping functions,J. Analyse Math. 52 (1989), 53–93.MathSciNetzbMATHCrossRefGoogle Scholar
  68. [68]
    S. SaitohTheory of reproducing kernels and its applicationsLongman Scientific & Technical, Harlow, 1988.zbMATHGoogle Scholar
  69. [69]
    L. SchwartzSous-espaces hilbertiens d’espaces vectoriels topologiques et noyaux associés (noyaux reproduisants)J. Analyse Math. 13 (1964), 115–256.MathSciNetzbMATHCrossRefGoogle Scholar
  70. [70]
    Yu.L. Shmul’yanDivision in the class of J-expansive operatorsMat. Sb. (N. S.) 74 (116) (1967), 516–525; Engl. transi.: Math. USSR-Sbornik 3 (1967), 471–479.zbMATHGoogle Scholar
  71. [71]
    P. SorjonenPontryaginräume mit einem reproduzierenden Kern,Ann. Acad. Sci. Fenn. Ser. A I Math. (1975), no. 594, 30 pages.MathSciNetGoogle Scholar
  72. [72]
    B.Sz.-Nagy and C. FoiasHarmonic analysis of operators on Hilbert spaceNorth-Holland Publishing Co., Amsterdam, 1970.Google Scholar
  73. [73]
    O. TammiExtremum problems for bounded univalent functions. I II Lecture Notes in Mathematics, Springer-Verlag, Berlin, vol. 646, 1978; ibid. vol. 913, 1982.Google Scholar
  74. [74]
    A.M. Yang, Aconstruction of unitary linear systemsIntegral Equations and Operator Theory 19 (1994), no. 4, 477–499.MathSciNetzbMATHCrossRefGoogle Scholar

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© Springer Basel AG 2002

Authors and Affiliations

  • James Rovnyak
    • 1
  1. 1.Department of MathematicsUniversity of VirginiaCharlottesvilleUSA

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