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The HHolomorphic Functional Calculus for Sectorial Operators — a Survey

  • Lutz Weis
Part of the Operator Theory: Advances and Applications book series (OT, volume 168)

Abstract

In this article we survey recent results on the holomorphic functional calculus for sectorial operators on Banach spaces. Starting from important classes of operators with a bounded H -calculus and its essential applications we show how the classical Hilbert space theory of the H -calculus can be extended to the Banach space setting. In particular, we discuss characterizations in terms of dilations, interpolation theory and square function estimates. These results lead to perturbation results which in turn allow us to verify the boundedness of the H -calculus for new classes of partial differential operators.

Keywords

Hilbert Space Banach Space Functional Calculus Sectorial Operator Convex Banach Space 
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References

  1. [1]
    D. Albrecht, Functional Calculi of Commuting Unbounded Operators, Ph.D. thesis, Monash University, Australia, 1994.Google Scholar
  2. [2]
    D. Albrecht, X. Duong and A. McIntosh, Operator theory and harmonic analysis, in: Instructional Workshop on Analysis and Geometry, Part III (Canberra, 1995), 77–136, Proc. Centre Math. Appl. Austral. Nat. Univ., vol. 34, Canberra, 1996.zbMATHMathSciNetGoogle Scholar
  3. [3]
    D. Albrecht, E. Franks, and A. McIntosh, Holomorphic functional calculi and sums of commuting operators, Bull. Austral. Math. Soc. 58 (1998), no. 2, 291–305.zbMATHMathSciNetGoogle Scholar
  4. [4]
    H. Amann, M. Hieber and G. Simonett, Bounded H -calculus for elliptic operators, Differential Integral Equations 7 (1994), no. 3–4, 613–653.zbMATHMathSciNetGoogle Scholar
  5. [5]
    W. Arendt, S. Bu, and M. Haase, Functional calculus, variational methods and Liapunov’s theorem, Arch. Math. 77 (2001), 65–75.zbMATHMathSciNetCrossRefGoogle Scholar
  6. [6]
    P. Auscher, S. Hofmann, M. Lacey, J. Lewis, A. McIntosh, and P. Tchamitchian, The solution of Kato’s conjectures, C. R. Acad. Sci. Paris Sér. I Math. 332 (2001), no. 7, 601–606.zbMATHMathSciNetGoogle Scholar
  7. [7]
    P. Auscher, S. Hofmann, A. McIntosh, and P. Tchamitchian, The Kato square root problem for higher order elliptic operators and systems onn, J. Evol. Equ. 1 (2001), no. 4, 361–385, Dedicated to the memory of Tosio Kato.zbMATHMathSciNetCrossRefGoogle Scholar
  8. [8]
    A. Axelson, S. Keith and A. McIntosh, Quadratic estimates and functional calculi of perturbed Dirac operators, preprint.Google Scholar
  9. [9]
    P. Auscher, A. McIntosh and A. Nahmod, Holomorphic functional calculi of operators, quadratic estimates and interpolation, Indiana Univ. Math. J. 46 (1997), no. 2, 375–403.zbMATHMathSciNetCrossRefGoogle Scholar
  10. [10]
    P. Auscher and P. Tchamitchian, Square root problem for divergence operators and related topics, Astérisque, no. 249, 1998.Google Scholar
  11. [11]
    J. Bergh and J. Löfström, Interpolation Spaces. An introduction, Springer-Verlag, Berlin, 1976, Grundlehren der Mathematischen Wissenschaften, vol. 223.Google Scholar
  12. [12]
    S. Blunck and P.C. Kunstmann, Calderon-Zygmund theory for non-integral operators and the H calculus, Rev. Math. Iberoamericana 19 (2003), no. 3, 919–942.zbMATHMathSciNetGoogle Scholar
  13. [13]
    K. Boyadzhiev and R. deLaubenfels, Spectral theorem for unbounded strongly continuous groups on a Hilbert space, Proc. Amer. Math. Soc. 120 (1994), no. 1, 127–136.zbMATHMathSciNetCrossRefGoogle Scholar
  14. [14]
    K. Boyadzhiev and R. deLaubenfels, Semigroups and resolvents of bounded variation, imaginary powers, and H -calculus, Semigroup Forum 45 (1992), 372–384.zbMATHMathSciNetGoogle Scholar
  15. [15]
    Ph. Clément, B. de Pagter, F.A. Sukochev, and H. Witvliet, Schauder decomposition and multiplier theorems, Studia Math. 138 (2000), no. 2, 135–163.zbMATHMathSciNetGoogle Scholar
  16. [16]
    Ph. Clément and J. Prüss, An operator-valued transference principle and maximal regularity on vector-valued L p-spaces, In: Evolution Equations and their Applications in Physical and Life Sciences (Bad Herrenalb, 1998), Dekker, New York, 2001,pp. 67–87.Google Scholar
  17. [17]
    M.G. Cowling, Harmonic analysis on semigroups, Ann. of Math. (2) 117 (1983), no. 2, 267–283.zbMATHMathSciNetCrossRefGoogle Scholar
  18. [18]
    M. Cowling, I. Doust, A. McIntosh, and A. Yagi, Banach space operators with a bounded H functional calculus, J. Austral. Math. Soc. Ser. A 60 (1996), no. 1, 51–89.zbMATHMathSciNetGoogle Scholar
  19. [19]
    G. Da Prato and P. Grisvard, Sommes d’opérateurs linéaires etéquations différentielles opérationnelles, J. Math. Pures Appl. (9) 54 (1975), no. 3, 305–387.zbMATHMathSciNetGoogle Scholar
  20. [20]
    R. Denk, M. Hieber and J. Prüss, R-boundedness, Fourier Multipliers and Problems of Elliptic and Parabolic Type, Mem. Amer. Math. Soc., vol. 166, Amer. Math. Soc, Providence, R.I., 2003.Google Scholar
  21. [21]
    R. Denk, G. Dore, M. Hieber, J. Prüss, and A. Venni, New thoughts on old resultsof R.T. Seeley, Math. Ann. 328 (2004), 545–583.zbMATHMathSciNetCrossRefGoogle Scholar
  22. [22]
    J. Dettweiler, J. van Neerven and L. Weis, Regularity of solutions of the stochastic abstract Cauchy problem, in preparation.Google Scholar
  23. [23]
    G. Dore, H functional calculus in real interpolation spaces II, Studia Math. 145 (2001), no. 1, 75–83.zbMATHMathSciNetCrossRefGoogle Scholar
  24. [24]
    G. Dore, H functional calculus in real interpolation spaces, Studia Math. 137 (1999), no. 2, 161–167.zbMATHMathSciNetGoogle Scholar
  25. [25]
    G. Dore, Fractional powers of closed operators, preprint.Google Scholar
  26. [26]
    G. Dore and A. Venni, On the closedness of the sum of two closed operators, Math. Z. 196 (1987), no. 2, 189–201.zbMATHMathSciNetCrossRefGoogle Scholar
  27. [27]
    G. Dore and A. Venni, Separation of two (possibly unbounded) components of the spectrum of a linear operator, Integral Equations and Operator Theory 12 (1989), 470–485.zbMATHMathSciNetCrossRefGoogle Scholar
  28. [28]
    M. Duelli, Functional Calculus for Bisectorial Operators and Applications to Linear and Lon-linear Evolution Equations, Ph.D. thesis, Universität Ulm, 2005.Google Scholar
  29. [29]
    M. Duelli and L. Weis, Spectral projections, Riesz transforms and H -calculus for bisectorial operators, to appear in Progress in Nonlinear Differential Equations and Their Applications, vol. 64, 2005.Google Scholar
  30. [30]
    N. Dunford and J.T. Schwartz, Linear Operators. Part III. Spectral Operators, Wiley Classics Library, John Wiley & Sons, Inc., New York, 1971.zbMATHGoogle Scholar
  31. [31]
    X. T. Duong and G. Simonett, H -calculus for elliptic operators with nonsmooth coefficients, Differential Integral Equations 10 (1997), no. 2, 201–217.zbMATHMathSciNetGoogle Scholar
  32. [32]
    X.T. Duong, H functional calculus of second-order elliptic partial differential operators on L p spaces, Miniconference on Operators in Analysis (Sydney, 1989), Austral. Nat. Univ., Canberra, 1990, pp. 91–102.Google Scholar
  33. [33]
    X.T. Duong, H -functional calculus of second order elliptic partial differential operators on L p-spaces, pp. 91–102, Proc. Centre for Math. Analysis Austral. Nat. Univ., Canberra, vol. 24, 1998.MathSciNetGoogle Scholar
  34. [34]
    X.T. Duong, Li Xi Yan, Bounded holomorphic functional calculus for non-divergence form operators, Differential Integral Equations 15 (2002), 709–730.zbMATHMathSciNetGoogle Scholar
  35. [35]
    X.T. Duong and A. McIntosh, Singular integral operators with non-smooth kernels on irregular domains, Rev. Mat. Iberoamericana 15 (1999), no. 2, 233–265.zbMATHMathSciNetGoogle Scholar
  36. [36]
    X. T. Duong and D.W. Robinson, Semigroup kernels, Poisson bounds, and holomorphic functional calculus, J. Funct. Anal. 142 (1996), no. 1, 89–128.zbMATHMathSciNetCrossRefGoogle Scholar
  37. [37]
    K.-J. Engel and R. Nagel, One-Parameter Semigroups for Linear Evolution Equations, Springer-Verlag, New York, 2000, With contributions by S. Brendle, M. Campiti, T. Hahn, G. Metafune, G. Nickel, D. Pallara, C. Perazzoli, A. Rhandi, S. Romanelli and R. Schnaubelt.zbMATHGoogle Scholar
  38. [38]
    G. Fendler, Dilations of one-parameter semigroups of positive contractions on L p-spaces, Canad. J. Math. 49 (1997), 736–748.zbMATHMathSciNetGoogle Scholar
  39. [39]
    S. Flory, F. Neubrander and L. Weis, Consistency and stabilization of rational approximation schemes for C 0-semigroups, in: Evolution Equations: Applications to Physics, Industry, Life Sciences and Economics, 181–193, G. Lumer, M. Ianelli, eds., Progr. Nonlinear Differential Equations Appl., vol. 55, Birkhäuser, Basel, 2003.Google Scholar
  40. [40]
    A. Fröhlich, Stokes-und Navier-Stokes-Gleichungen in Gewichteten Funktionenräumen, Ph.D. thesis, TU Darmstadt, 2001, Shaker Verlag.Google Scholar
  41. [41]
    A.M. Fröhlich and L. Weis, H -calculus of sectorial operators and dilations, submitted.Google Scholar
  42. [42]
    Y. Giga, Domains of fractional powers of the Stokes operator in L r spaces, Arch. Rational Mech. Anal. 89 (1985), no. 3, 251–265.zbMATHMathSciNetCrossRefGoogle Scholar
  43. [43]
    B.H. Haak, Kontrolltheorie in Banachräumen und Quadratische Abschätzungen, Universitätsverlag Karlsruhe, 2005, http://www.ubka.uni-karlsruhe.de/vvv/2004/ mathematik/3/3.pdfGoogle Scholar
  44. [44]
    B.H. Haak, C. Le Merdy, α-Admissibility of control and observation operators, to appear in Houston J. Math.Google Scholar
  45. [45]
    M. Haase, The Functional Calculus for Sectorial Operators and Similarity Methods, Ph.D. thesis, Universität Ulm, 2003.Google Scholar
  46. [46]
    M. Haase, Spectral mapping theorems for functional calculi, Ulmer Seminare in Funktionalanalysis und Differentialgleichungen 2003, 209–224.Google Scholar
  47. [47]
    M. Haase, A characterization of group operators on Hilbert space and functional calculus, Semigroup Forum 66 (2003), no. 2, 288–304.zbMATHMathSciNetCrossRefGoogle Scholar
  48. [48]
    M. Haase, Spectral properties of operator logarithms, Math. Z. 245 (2003), no.4, 761–779.zbMATHMathSciNetCrossRefGoogle Scholar
  49. [49]
    H. Heck, M. Hieber, Maximal L p-regularity for elliptic operators with VMO-coefficients, J. Evol. Equ. 3 (2003) no. 2, 332–359.zbMATHMathSciNetGoogle Scholar
  50. [50]
    M. Hieber and J. Prü Functional calculi for linear operators in vector-valued L p-spaces via the transference principle, Adv. Differential Equations 3 (1998), no. 6, 847–872.zbMATHMathSciNetGoogle Scholar
  51. [51]
    M. Hoffmann, N. Kalton, T. Kucherenko: R-bounded approximating sequences and applications to semigroups, J. Math. Anal. Appl. 294 (2002) no. 2., 373–386.MathSciNetCrossRefGoogle Scholar
  52. [52]
    M. Junge, C. Le Merdy and Q. Xu, Calcul fonctionnel et fonctions carrées dans les espaces L p non commutatifs, C. R. Math. Acad. Sci. Paris 337 (2003), no. 2, 93–98.zbMATHMathSciNetGoogle Scholar
  53. [53]
    N.J. Kalton, A remark on sectorial operators with an H -calculus, Trends in Banach Spaces and Operator Theory, pp. 91–99, Contemp. Math., vol. 321, AMS, Rhode Island, 2003.Google Scholar
  54. [54]
    N.J. Kalton, P.C. Kunstmann and L. Weis, Perturbation and interpolation theorems for the H -calculus with applications to differential operators, to appear in Math. Ann.Google Scholar
  55. [55]
    N.J. Kalton and L. Weis, The H -calculus and sums of closed operators, Math. Ann. 321 (2001), no. 2, 319–345.zbMATHMathSciNetCrossRefGoogle Scholar
  56. [56]
    N.J. Kalton and L. Weis, H -functional calculus and square functions estimates, in preparation.Google Scholar
  57. [57]
    N.J. Kalton and L. Weis, Euclidean structures and their applications to spectral theory, in preparation.Google Scholar
  58. [58]
    H. Komatsu, Fractional powers of operators, Pac. J. Math. 19 (1966), 285–346.zbMATHMathSciNetGoogle Scholar
  59. [59]
    T. Kucherenko and L. Weis, Real interpolation of domains of sectorial operators on L p-spaces, submitted.Google Scholar
  60. [60]
    P.C. Kunstmann and L. Weis, Perturbation theorems for maximal L p-regularity, Ann. Sc. Norm. Sup. Pisa XXX (2001), 415–435.MathSciNetGoogle Scholar
  61. [61]
    P.C. Kunstmann and L. Weis, Maximal L p-regularity for Parabolic equations, Fourier multiplier theorems and H -functional calculus, in: Functional Analytic Methods for Evolution Equations, M. Ianelli, R. Nagel and S. Piazzera, eds., Lecture Notes in Mathematics, vol. 1855, pp. 65–311, Springer, 2004.Google Scholar
  62. [62]
    F. Lancien, G. Lancien, and C. Le Merdy, A joint functional calculus for sectorial operators with commuting resolvents, Proc. London Math. Soc. (3) 77 (1998), no. 2, 387–414.zbMATHMathSciNetCrossRefGoogle Scholar
  63. [63]
    S. Larsson, V. Thomée and L.B. Wahlbin, Finite-element methods for a strongly damped wave equation, IMA J. Numer. Anal. 11 (1991), no. 1, 115–142.zbMATHMathSciNetGoogle Scholar
  64. [64]
    C. Le Merdy, H -functional calculus and applications to maximal regularity, Publ. Math. UFR Sci. Tech. Besançon. 16 (1998), 41–77.zbMATHGoogle Scholar
  65. [65]
    C. Le Merdy, The similarity problem for bounded analytic semigroups on Hilbert space, Semigroup Forum 56 (1998), 205–224.zbMATHMathSciNetCrossRefGoogle Scholar
  66. [66]
    C. Le Merdy, Similarities of ω-accretive operators, Internat. Conf. on Harmonic Analysis and Related Topics (Sydney 2002), 85–94 Proc. Centre Math. Appl. Austral. Nat. Univ., vol. 41, Canberra, 2003.Google Scholar
  67. [67]
    C. Le Merdy, On square functions associated to sectorial operators, Bull. Soc. Math. France 132 (2004), pp. 137–156.zbMATHMathSciNetGoogle Scholar
  68. [68]
    C. Le Merdy, Two results about H functional calculus on analytic UMD Banach spaces, J. Austral. Math. Soc. 74 (2003), pp. 351–378.zbMATHCrossRefGoogle Scholar
  69. [69]
    C. Le Merdy, The Weiss conjecture for bounded analytic semigroups, J. London Math. Soc. 67, (2003), no. 3, 715–738.zbMATHMathSciNetCrossRefGoogle Scholar
  70. [70]
    A. McIntosh, Operators which have an H functional calculus, Miniconference on Operator Theory and Partial Differential Equations (North Ryde, 1986), Austral. Nat. Univ., Canberra, 1986, pp. 210–231.Google Scholar
  71. [71]
    A. McIntosh, On representing closed accretive sesquilinear forms as (A 1/2 u, A*1/2 v), Nonlinear Partial Differential Equations and their Applications, Collège de France Seminar, Vol. III (Paris, 1980/1981), pp. 252–267, Res. Notes in Math., vol. 70, Pitman, Boston, Mass.-London, 1982.Google Scholar
  72. [72]
    A. McIntosh and A. Yagi, Operators of type ω without a bounded H -functional calculus, Proc. Cent. Math. Anal. Aust. Natl. Univ. 24 (1990), 159–172.zbMATHMathSciNetGoogle Scholar
  73. [73]
    A. Noll and J. Saal, H -calculus for the Stokes operator on L q-spaces, Math. Z. 244 (2003) no. 3, 651–688.zbMATHMathSciNetGoogle Scholar
  74. [74]
    A. Pazy, Semigroups of linear operators and applications to partial differential equations, Springer-Verlag, New York, 1983.zbMATHGoogle Scholar
  75. [75]
    J. Prüss and G. Simonett, H -calculus for the sum of non-commuting operators, preprint.Google Scholar
  76. [76]
    R.T. Seeley, Complex powers of an elliptic operator, Singular Integrals (Proc. Sympos. Pure Math., Chicago, Ill., 1966), Amer. Math. Soc., Providence, R.I., 1967, pp. 288–307.Google Scholar
  77. [77]
    E.M. Stein, Singular Integrals and Differentiability Properties of Functions, Princeton University Press, 1970.Google Scholar
  78. [78]
    E.M. Stein, Topics in Harmonic Analysis Related to the Littlewood-Paley Theory, Princeton University Press and the University of Tokyo Press, 1970.Google Scholar
  79. [79]
    M. Uiterdijk, A functional calculus for analytic generators of C 0-groups, Integral Equations Operator Theory 36 (2000), no. 3, 349–369.zbMATHMathSciNetCrossRefGoogle Scholar
  80. [80]
    L. Weis, A new approach to maximal L p-regularity, Evolution Equations and their Applications in Physical and Life Sciences (Bad Herrenalb, 1998), Dekker, New York, 2001, pp. 195–214.Google Scholar
  81. [81]
    L. Weis, Operator-valued Fourier multiplier theorems and maximal L p-regularity, Math. Ann. 319 (2001), no. 4, 735–758.zbMATHMathSciNetCrossRefGoogle Scholar
  82. [82]
    A. Yagi, Coϊncidence entre des espaces d’interpolation et des domaines de puissances fractionnaires d’opérateurs C. R. Acad. Sci., Paris, Sér. I 299 (1984), 173–176.zbMATHMathSciNetGoogle Scholar
  83. [83]
    A. Yagi, Applications of the purely imaginary powers of operators in Hilbert spaces, J. Funct. Anal. 73 (1987), no. 1, 216–231.zbMATHMathSciNetCrossRefGoogle Scholar

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© Birkhäuser Verlag Basel/Switzerland 2006

Authors and Affiliations

  • Lutz Weis
    • 1
  1. 1.Mathematisches Institut ITechnische Universität KarlsruheKarlsruheGermany

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