Open Access

Infinitesimal generators of invertible evolution families


A logarithm representation of operators is introduced as well as a concept of pre-infinitesimal generator. Generators of invertible evolution families are represented by the logarithm representation, and a set of operators represented by the logarithm is shown to be associated with analytic semigroups. Consequently generally-unbounded infinitesimal generators of invertible evolution families are characterized by a convergent power series representation.

Key words: Invertible evolution family, operator theory, maximal regularity.

Full Text

Article Information

TitleInfinitesimal generators of invertible evolution families
SourceMethods Funct. Anal. Topology, Vol. 23 (2017), no. 1, 26–36
MilestonesReceived 31/10/2016; Revised 17/12/2016
CopyrightThe Author(s) 2017 (CC BY-SA)

Authors Information

Yoritaka Iwata
Institute of Innovative Research, Tokyo Institute of Technology; Department of Mathematics, Shibaura Institute of Technology

Google Scholar Metrics

Citing articles in Google Scholar
Similar articles in Google Scholar

Export article

Save to Mendeley

Citation Example

Yoritaka Iwata, Infinitesimal generators of invertible evolution families, Methods Funct. Anal. Topology 23 (2017), no. 1, 26–36.


@article {MFAT944,
    AUTHOR = {Iwata, Yoritaka},
     TITLE = {Infinitesimal generators of invertible evolution families},
   JOURNAL = {Methods Funct. Anal. Topology},
  FJOURNAL = {Methods of Functional Analysis and Topology},
    VOLUME = {23},
      YEAR = {2017},
    NUMBER = {1},
     PAGES = {26–36},
      ISSN = {1029-3531},
       URL = {},


  1. Wolfgang Arendt, Semigroups and evolution equations: functional calculus, regularity and kernel estimates, Evolutionary equations. Vol. I, Handb. Differ. Equ., North-Holland, Amsterdam, 2004, pp. 1-85.  MathSciNet
  2. Khristo N. Boyadzhiev, Logarithms and imaginary powers of operators on Hilbert spaces, Collect. Math. 45 (1994), no. 3, 287-300.  MathSciNet
  3. Nelson Dunford, Spectral theory. I. Convergence to projections, Trans. Amer. Math. Soc. 54 (1943), 185-217.  MathSciNet
  4. Markus Haase, Spectral properties of operator logarithms, Math. Z. 245 (2003), no. 4, 761-779.  MathSciNet CrossRef
  5. Markus Haase, The functional calculus for sectorial operators, Operator Theory: Advances and Applications, vol. 169, Birkhauser Verlag, Basel, 2006.  MathSciNet CrossRef
  6. Tosio Kato, Perturbation theory for linear operators, Die Grundlehren der mathematischen Wissenschaften, Band 132, Springer-Verlag New York, Inc., New York, 1966.  MathSciNet
  7. Tosio Kato, Linear evolution equations of ``hyperbolic type, J. Fac. Sci. Univ. Tokyo Sect. I 17 (1970), 241-258.  MathSciNet
  8. Tosio Kato, Linear evolution equations of ``hyperbolic type. II, J. Math. Soc. Japan 25 (1973), 648-666.  MathSciNet
  9. Tosio Kato, A short introduction to perturbation theory for linear operators, Springer-Verlag, New York-Berlin, 1982.  MathSciNet
  10. S. G. Krein, Linear differential equations in Banach space, American Mathematical Society, Providence, R.I., 1971.  MathSciNet
  11. Celso Mart\inez Carracedo and Miguel Sanz Alix, The theory of fractional powers of operators, North-Holland Mathematics Studies, vol. 187, North-Holland Publishing Co., Amsterdam, 2001.  MathSciNet
  12. Volker Nollau, Uber den Logarithmus abgeschlossener Operatoren in Banachschen Raumen, Acta Sci. Math. (Szeged) 30 (1969), 161-174.  MathSciNet
  13. Noboru Okazawa, Logarithms and imaginary powers of closed linear operators, Integral Equations Operator Theory 38 (2000), no. 4, 458-500.  MathSciNet CrossRef
  14. Noboru Okazawa, Logarithmic characterization of bounded imaginary powers, Semigroups of operators: theory and applications (Newport Beach, CA, 1998), Progr. Nonlinear Differential Equations Appl., vol. 42, Birkhauser, Basel, 2000, pp. 229-237.  MathSciNet
  15. A. Pazy, Semigroups of linear operators and applications to partial differential equations, Applied Mathematical Sciences, vol. 44, Springer-Verlag, New York, 1983.  MathSciNet CrossRef
  16. Jan Pruss and Roland Schnaubelt, Solvability and maximal regularity of parabolic evolution equations with coefficients continuous in time, J. Math. Anal. Appl. 256 (2001), no. 2, 405-430.  MathSciNet CrossRef
  17. Hiroki Tanabe, Equations of evolution, Monographs and Studies in Mathematics, vol. 6, Pitman (Advanced Publishing Program), Boston, Mass.-London, 1979.  MathSciNet
  18. Angus E. Taylor, Spectral theory of closed distributive operators, Acta Math. 84 (1951), 189-224.  MathSciNet

All Issues