, Volume 7, Issue 3, pp 195–202 | Cite as

Operator Semigroups for which Reducibility Implies Decomposability

  • L. Livshits
  • G. Macdonald
  • B. Mathes
  • H. Radjavi


A description of the lattice of invariant subspaces is provided for multiplicative semigroups \(\mathcal{S}\) of bounded operators on Lp(X,μ) which are closed under multiplication on the left or right by bounded multiplication operators. Applications are then given to semigroups of positive quasinilpotent operators.

decomposable positive operator quasinilpotent reducible semigroup 


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  1. 1.
    Y.A. Abramovich, C.D. Aliprantis and O. Burkinshaw: On the spectral radius of positive operators, Math. Z. 211 (1992), 593–607. Corrigendum: Math. Z. 215 (1994), 167–168.Google Scholar
  2. 2.
    Y.A. Abramovich, C.D. Aliprantis and O. Burkinshaw: Invariant subspaces of operators on p spaces, J. Funct. Anal. 115 (1993), 418–424.Google Scholar
  3. 3.
    Y.A. Abramovich, C.D. Aliprantis and O. Burkinshaw: Invariant subspaces for positive operators, J. Funct. Anal. 124 (1994), 95–111.Google Scholar
  4. 4.
    T. Ando: Positive operators in semi-ordered linear spaces, J. Fac. Sci. Hokkaido Univ., Ser. 1 13 (1957), 214–228.Google Scholar
  5. 5.
    R. Drnovšek: Common invariant subspaces for collections of operators, to appear in Journal of Integral Equations and Operator Theory.Google Scholar
  6. 6.
    P. Halmos: A HilbertSpace Problem Book, Springer, New York, 1982.Google Scholar
  7. 7.
    H.J. Krieger: Beitrage zur Theorie positiver Operatoren, Schriftenreihe der Institute fur Math. Reihe A, Heft 6; Akad.-Verlag, Berlin, 1969.Google Scholar
  8. 8.
    G. MacDonald: Invariant subspaces for Bishop-type operators, J. Funct. Anal. 91(2) (1990) 287–311.Google Scholar
  9. 9.
    B. de Pagter: Irreducible compact operators, Math. Z. 192 (1986) 149–153.Google Scholar
  10. 10.
    H. Radjavi and P. Rosenthal: Simultaneous Triangularization, Universitext, Springer, New York, 2000.Google Scholar
  11. 11.
    H. Schaefer: Banach lattices and positive operators, Springer, New York, 1974.Google Scholar
  12. 12.
    Y.V. Turovski202-1i: Volterra operators have invariant subspaces, J. Funct. Anal. 162(2) (1999) 313–322.Google Scholar
  13. 13.
    W.A.J. Luxemburg and A.C. Zaanen: Riesz Spaces I, North-Holland, Amsterdam, 1971.Google Scholar
  14. 14.
    A.C. Zaanen: Riesz Spaces II, North-Holland, Amsterdam, 1983.Google Scholar

Copyright information

© Kluwer Academic Publishers 2003

Authors and Affiliations

  • L. Livshits
    • 1
  • G. Macdonald
    • 2
  • B. Mathes
    • 1
  • H. Radjavi
    • 3
  1. 1.Department of MathematicsColby CollegeWatervilleUSA
  2. 2.Department of Math and CSUniversity of Prince Edward IslandCharlottetownCanada
  3. 3.Department of Math, Stats and CSDalhousie UniversityHalifaxCanada

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