Functional Analysis and Its Applications

, Volume 46, Issue 3, pp 210–217 | Cite as

A joint spectral mapping theorem for sets of semigroup generators



In the context of the multidimensional functional calculus of semigroup generators, which is based on the class of Bernstein functions in several variables (and is also known as Bochner-Phillips multidimensional functional calculus), a spectral mapping theorem for the Taylor spectrum of a set of commuting generators is proved.

Key words

multiparameter semigroup of operators multidimensional functional calculus Taylor spectrum Bernstein function spectral mapping theorem 


Unable to display preview. Download preview PDF.

Unable to display preview. Download preview PDF.


  1. [1]
    A. Kishimoto and D. Robinson, “Subordinate semigroups and order properties,” J. Austral. Math. Soc. Ser. A, 31:1 (1981), 59–76.MathSciNetMATHCrossRefGoogle Scholar
  2. [2]
    Ch. Berg, K. Boyadzhiev, and R. deLaubenfels, “Generation of generators of holomorphic semigroups,” J. Austral. Math. Soc. Ser. A, 55:2 (1993), 246–269.MathSciNetMATHCrossRefGoogle Scholar
  3. [3]
    A. S. Carasso and T. Kato, “On subordinated holomorphic semigroups,” Trans. Amer. Math. Soc., 327:2 (1991), 867–878.MathSciNetMATHCrossRefGoogle Scholar
  4. [4]
    A. R. Mirotin, “On the T-calculus of generators for C 0-semigroups,” Sibirsk. Mat. Zh., 39:3 (1998), 571–582; English transl.: Siberian Math. J., 39:3 (1998), 493–503.MathSciNetMATHGoogle Scholar
  5. [5]
    R. Shilling, R. Song, and Z. Vondraček, Bernstein Functions. Theory and Applications, de Greyter, Berlin-New York, 2010.Google Scholar
  6. [6]
    W. Feller, An Introduction to Probability Theory and Its Applications, vol. 2, Wiley, New York-London-Sidney, 1971.Google Scholar
  7. [7]
    E. Hille and R. Phillips, Functional Analysis and Semigroups, Amer. Math. Soc., Providence, RI, 1957.MATHGoogle Scholar
  8. [8]
    D. Applebaum, “Levy processes—from probability to finance and quantum groups,” Notices Amer. Math. Soc., 51:11 (2004), 1336–1347.MathSciNetMATHGoogle Scholar
  9. [9]
    A. R. Mirotin, “Functions from the Schoenberg class T on the cone of dissipative elements of a Banach algebra,” Mat. Zametki, 61:4 (1997), 630–633; English transl.: Math. Notes, 61:4 (1997), 524–527.MathSciNetCrossRefGoogle Scholar
  10. [10]
    A. R. Mirotin, “Functions from the Schoenberg class T act in the cone of dissipative elements of a Banach algebra, II,” Mat. Zametki, 64:3 (1998), 423–430; English transl.: Math. Notes, 64:3 (1998), 364–370.MathSciNetCrossRefGoogle Scholar
  11. [11]
    A. R. Mirotin, “Multidimensional T -calculus of generators of C 0 semigroups,” Algebra i Analiz, 11:2 (1999), 142–170; English transl.: St. Petersburg Math. J., 11:2 (2000), 315–335.MathSciNetGoogle Scholar
  12. [12]
    Ch. Berg, J. P. R. Christensen, and P. Ressel, Harmonic Analysis on Semigroups, Graduate Texts in Math., vol. 100, Springer-Verlag, New York-Berlin, 1984.Google Scholar
  13. [13]
    A. R. Mirotin, “On multidimensional Bochner-Phillips functional calculus,” Prob. Fiz. Mat. Tekh., 1:1 (2009), 63–66.Google Scholar
  14. [14]
    O. Bratelli and D. W. Robinson, Operator Algebras and Quantum Statistical Mechanics, Springer-Verlag, New York, 1979.Google Scholar
  15. [15]
    J. A. Goldstein, Semigroups of Linear Operators and Applications, Clarendon Press, Oxford University Press, New York, 1985.MATHGoogle Scholar
  16. [16]
    A. Ya. Khelemskii, Homology in Banach and Topological Algebras [in Russian], Izd. Mosk. Gos. Univ., Moscow, 1986.Google Scholar
  17. [17]
    Z. Slodkowski and W. Zelazko, “On joint spectra of commuting families of operators,” Studia Math., 50 (1974), 127–148.MathSciNetMATHGoogle Scholar
  18. [18]
    Ph. Clément, H. Heijmans, S. Angenent, et al., One-Parameter Semigroups, North-Holland, Amsterdam, 1987.MATHGoogle Scholar
  19. [19]
    T. Kato, Perturbation Theory for Linear Operators, 2nd ed., Springer-Verlag, Berlin, 1976.MATHCrossRefGoogle Scholar
  20. [20]
    A. Ya. Khelemskii, “Homological methods in Taylor’s holomorphic calculus of several operators in a Banach space,” Uspekhi Mat. Nauk, 36:(217) (1981), 127–172; English transl.: Russian Math. Surveys, 36:1 (1981), 139–192.MathSciNetGoogle Scholar
  21. [21]
    J. L. Taylor, “A joint spectrum for several commuting operators,” J. Funct. Anal., 6:2 (1970), 172–191.MATHCrossRefGoogle Scholar

Copyright information

© Springer-Verlag 2012

Authors and Affiliations

  1. 1.F. Skorina Gomel State UniversityGomelRussia

Personalised recommendations