Integral Equations and Operator Theory

, Volume 38, Issue 4, pp 458–500 | Cite as

Logarithms and imaginary powers of closed linear operators

  • Noboru Okazawa
Article

Abstract

The imaginary powersAit of a closed linear operatorA, with inverse, in a Banach spaceX are considered as aC0-group {exp(itlogA);tR} of bounded linear operators onX, with generatori logA. Here logA is defined as the closure of log(1+A) − log(1+A−1). LetA be a linearm-sectorial operator of typeS(tan ω), 0≤ω≤(π/2), in a Hilbert spaceX. That is, |Im(Au, u)| ≤ (tan ω)Re(Au, u) foruD(A). Then ω±ilog(1+A) ism-accretive inX andilog(1+A) is the generator of aC0-group {(1+A)it;tR} of bounded imaginary powers, satisfying the estimate ‖(1+A)it‖ ≤ exp(ω|t|),tR. In particular, ifA is invertible, then ω±ilogA ism-accretive inX, where logA is exactly given by logA=log(1+A)−log(1+A−1), and {Ait;tR} forms aC0-group onX, with the estimate ‖Ait‖ ≤ exp(ω|t|),tR. This yields a slight improvement of the Heinz-Kato inequality.

MSC 1991

Primary 47B44 Secondary 47D03 

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References

  1. 1.
    H. Amann,Linear and Quasilinear Parabolic Problems, I:Abstract Linear Theory, Monographs in Math., vol. 89, Birkhäuser Verlag, Basel and Boston, 1995.Google Scholar
  2. 2.
    W. Arendt, O. El Mennaoui and M. Hieber,Boundary values of holomorphic semigroups, Proc. Amer. Math. Soc.125 (1997), 635–647.Google Scholar
  3. 3.
    J.B. Baillon and Ph. Clément,Examples of unbounded imaginary powers of operators, J. Functional Anal.100 (1991), 419–434.Google Scholar
  4. 4.
    W. Borchers and T. Miyakawa,L 2 decay for the Navier-Stokes flow in halfspaces, Math. Ann.282 (1988), 139–155.Google Scholar
  5. 5.
    H. Brezis,Quelques propriétés des opérateurs monotones et des semi-groupes non linéaires, Lecture Notes in Math., vol. 543, Springer-Verlag, Berlin and New York, 1976, pp. 56–82.Google Scholar
  6. 6.
    H. Brezis, M.G. Crandall and A. Pazy,Perturbations of nonlinear maximal monotone sets in Banach space, Comm. Pure Appl. Math.23 (1970), 123–144.Google Scholar
  7. 7.
    A.S. Carasso and T. Kato,On subordinated holomorphic semigroups, Trans. Amer. Math. Soc.327 (1991), 867–878.Google Scholar
  8. 8.
    R.R. Coifman and G. Weiss,Transference Method in Analysis, CBMS Regional Conference Ser. in Math. No. 31, Amer. Math. Soc., Providence, Rhode Island, 1977.Google Scholar
  9. 9.
    E.B. Davies,One-Parameter Semigroups, London Mathematical Society Monographs, vol. 15, Academic Press, London and New York, 1980.Google Scholar
  10. 10.
    G. Dore,L p regularity for abstract differential equations, Lecture Notes in Math., vol. 1540, Springer-Verlag, Berlin and New York, 1993, pp. 25–38.Google Scholar
  11. 11.
    G. Dore and A. Venni,On the closedness of the sum of two closed operators, Math. Z.196 (1987), 189–201.Google Scholar
  12. 12.
    H.O. Fattorini,The Cauchy Problem, Encyclopedia of Math. Appl., vol. 18, Cambridge Univ. Press (Addison-Wesley Publishing Com.), New York (London), 1984 (1983).Google Scholar
  13. 13.
    Y. Giga and H. Sohr,Abstract L p estimates for the Cauchy problem with applications to the Navier-Stokes equations in exterior domains, J. Functional Anal.102 (1991), 72–94.Google Scholar
  14. 14.
    J. Glimm and A. Jaffe,Singular perturbations of selfadjoint operators, Comm. Pure Appl. Math.22 (1969), 401–414.Google Scholar
  15. 15.
    J.A. Goldstein,Semigroups of Linear Operators and Applications, Oxford University Press, New York, 1985.Google Scholar
  16. 16.
    E. Heinz,Beiträge zur Störungstheorie der Spektralzerlegung, Math. Ann.123 (1951), 415–438.Google Scholar
  17. 17.
    M. Hieber, A. Holderrieth and F. Neubrander,Regularized semigroups and systems of linear partial differential equations, Ann. Scuola Norm. Sup. Pisa Cl. Sci. (4)19 (1992), 363–379.Google Scholar
  18. 18.
    E. Hille and R.S. Phillips,Functional Analysis and Semi-groups, Colloquium Publications, vol. 31, Amer. Math. Soc., Providence, Rhode Island, 1957.Google Scholar
  19. 19.
    T. Kato,Remarks on pseudo-resolvents and infinitesimal generators of semigroups, Proc. Japan Acad.35 (1959), 467–468.Google Scholar
  20. 20.
    T. Kato,Fractional powers of dissipative operators, J. Math. Soc. Japan13 (1961), 246–274.Google Scholar
  21. 21.
    T. Kato,Fractional powers of dissipative operators, II, J. Math. Soc. Japan14 (1962), 242–248.Google Scholar
  22. 22.
    T. Kato,A Generalization of the Heinz inequality, Proc. Japan Acad.37 (1961), 305–308.Google Scholar
  23. 23.
    T. Kato,Perturbation Theory for Linear Operators, Grundlehren Math. Wiss., vol. 132, Springer-Verlag, Berlin and New York, 1966; 2nd ed., 1976.Google Scholar
  24. 24.
    H. Komatsu,Fractional powers of operators, Pacific J. Math.19 (1966), 285–346.Google Scholar
  25. 25.
    H. Komatsu,Fractional powers of operators, VI:Interpolation of non-negative operators and imbeding theorems, J. Fac. Sci. Univ. Tokyo, Sec. IA19 (1972), 1–63.Google Scholar
  26. 26.
    S.G. Krein,Linear Differential Equations in Banach Space, Translations of Math. Mono., vol. 29, Amer. Math. Soc., Providence, Rhode Island, 1971.Google Scholar
  27. 27.
    H. Langer,Über die Wurzeln eines Maximalen Dissipativen Operatoren, Acta Math. Acad. Sci. Hungary13 (1962), 415–424.Google Scholar
  28. 28.
    C. Le Merdy,The similarity problem for bounded analytic semigroups on Hilbert space, Semigroup Forum56 (1998), 205–224.Google Scholar
  29. 29.
    V.A. Liskevich and M.A. Perelmuter,Analyticity of submarkovian semigroups, Proc. Amer. Math. Soc.123 (1995), 1097–1104.Google Scholar
  30. 30.
    C. Martinez, M. Sanz and L. Marco,Fractional powers of operators, J. Math. Soc. Japan40 (1988), 331–347.Google Scholar
  31. 31.
    A. McIntosh,Operators which have an H functional calculus, Miniconference on Operator Theory and Partial Differential Equations, 1986. Proceedings of the Centre for Mathematical Analysis, vol. 14, Australian National University, Canberra, 1986, pp. 210–231.Google Scholar
  32. 32.
    A. McIntosh and A. Yagi,Operators of type ω without a bounded H -functional calculus, Miniconference on Operators in Analysis, 1989. Proceedings of the Centre for Mathematical Analysis, vol. 24, Australian National University, Canberra, 1990, pp. 159–172.Google Scholar
  33. 33.
    S. Monniaux,A perturbation result for bounded imaginary powers, Arch. Math.68 (1997), 407–417.Google Scholar
  34. 34.
    S. Monniaux and J. Prüss,A theorem of the Dore-Venni type for noncommuting operators, Trans. Amer. Math. Soc.349 (1997), 4787–4814.Google Scholar
  35. 35.
    T. Muramatu,Interpolation Spaces and Linear Operators, Kinokuniya, Tokyo, 1985.Google Scholar
  36. 36.
    V. Nollau,Über den Logarithmus abgeschlossener Operatoren in Banachschen Räumen, Acta Sci. Math.30 (1969), 161–174.Google Scholar
  37. 37.
    N. Okazawa,Singular perturbations of m-accretive operators, J. Math. Soc. Japan32 (1980), 19–44.Google Scholar
  38. 38.
    N. Okazawa,Sectorialness of second order elliptic operators in divergence form, Proc. Amer. Math. Soc.113 (1991), 701–706.Google Scholar
  39. 39.
    N. Okazawa,L p-theory of Schrödinger operators with strongly singular potentials, Japanese J. Math.22 (1996), 199–239.Google Scholar
  40. 40.
    A. Pazy,Semigroups of Linear Operators and Applications to Partial Differential Equations, Applied Math. Sciences, vol. 44, Springer-Verlag, Berlin and New York, 1983.Google Scholar
  41. 41.
    J. Prüss,Evolutionary Integral Equations and Applications, Mono. in Math., vol. 87, Birkhäuser Verlag, Basel and Boston, 1993.Google Scholar
  42. 42.
    J. Prüss and H. Sohr,On operators with bounded imaginary powers in Banach spaces, Math. Z.203 (1990), 429–452.Google Scholar
  43. 43.
    J. Prüss and H. Sohr,Imaginary powers of elliptic second order differential operators in L p-spaces, Hiroshima Math. J.23 (1993), 161–192.Google Scholar
  44. 44.
    B. Sz.-Nagy and C. Foias,Harmonic Analysis of Operators on Hilbert Space, North-Holland, Amsterdam, 1970.Google Scholar
  45. 45.
    H. Tanabe,Equations of Evolution, Monographs and Studies in Math., vol. 6, Pitman, London and San Francisco, 1979.Google Scholar
  46. 46.
    N. Tanaka,On the exponentially bounded C-semigroups, Tokyo J. Math.10 (1987), 107–117.Google Scholar
  47. 47.
    H. Triebel,Interpolation Theory, Function Spaces, Differential Operators, North-Holland, Amsterdam, 1978.Google Scholar
  48. 48.
    A. Venni,A counterexample concerning imaginary powers of linear operators, Lecture Notes in Math., vol. 1540, Springer-Verlag, Berlin and New York, 1993, pp. 381–387.Google Scholar
  49. 49.
    A. Yagi,Applications of the purely imaginary powers of operators in Hilbert spaces, J. Functional Anal.73 (1987), 216–231.Google Scholar
  50. 50.
    A. Yoshikawa,On the logarithm of closed linear operators, Proc. Japan Acad.49 (1973), 169–173.Google Scholar
  51. 51.
    K. Yosida,Ergodic theorems for pseudo-resolvents, Proc. Japan Acad.37 (1961), 422–425.Google Scholar
  52. 52.
    K. Yosida,Functional Analysis, Grundlehren Math. Wiss., vol. 123, Springer-Verlag, Berlin and New York, 1965; 6th ed., 1980.Google Scholar

Copyright information

© Birkhäuser Verlag 2000

Authors and Affiliations

  • Noboru Okazawa
    • 1
  1. 1.Department of MathematicsScience University of TokyoTokyoJapan

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