Results in Mathematics

, 74:174 | Cite as

Counterexamples Related to Commutators of Unbounded Operators

  • Mohammed Hichem MortadEmail author


The present paper is exclusively devoted to examples and counterexamples about commutators and self commutators of unbounded operators on a Hilbert space. As a bonus, we provide a simpler counterexample than McIntosh’s famous example obtained some while ago.


Commutators self-commutators bounded and unbounded operators 

Mathematics Subject Classification

Primary 47B47 Secondary 47A05 47B25 



The author wishes to thank the referee for all his/her remarks and suggestions which have been necessary to improve some parts of the paper.


  1. 1.
    Dehimi, S., Mortad, M.H.: Chernoff like counterexamples related to unbounded operators. Kyushu J. Math. (to appear)Google Scholar
  2. 2.
    Fong, C.K.: Norm estimates related to self-commutators. Linear Algebra Appl. 74, 151–156 (1986)MathSciNetCrossRefGoogle Scholar
  3. 3.
    Kittaneh, F.: Inequalities for commutators of positive operators. J. Funct. Anal. 250(1), 132–143 (2007)MathSciNetCrossRefGoogle Scholar
  4. 4.
    Kittaneh, F.: Norm inequalities for commutators of positive operators and applications. Math. Z 258(4), 845–849 (2008)MathSciNetCrossRefGoogle Scholar
  5. 5.
    Kittaneh, F.: Norm inequalities for commutators of self-adjoint operators. Integral Equ. Oper. Theory 62(1), 129–135 (2008)MathSciNetCrossRefGoogle Scholar
  6. 6.
    Kosaki, H.: On intersections of domains of unbounded positive operators. Kyushu J. Math. 60(1), 3–25 (2006)MathSciNetCrossRefGoogle Scholar
  7. 7.
    McIntosh, A.: Counterexample to a question on commutators. Proc. Am. Math. Soc. 29, 337–340 (1971)MathSciNetCrossRefGoogle Scholar
  8. 8.
    Möller, M., Szafraniec, F.H.: Adjoints and formal adjoints of matrices of unbounded operators. Proc. Am. Math. Soc. 136(6), 2165–2176 (2008)MathSciNetCrossRefGoogle Scholar
  9. 9.
    Mortad, M.H.: On the triviality of domains of powers and adjoints of closed operators. Acta Sci. Math. (Szeged) (to appear)Google Scholar
  10. 10.
    Nelson, E.: Analytic vectors. Ann. Math. 70, 572–615 (1959)MathSciNetCrossRefGoogle Scholar
  11. 11.
    Ôta, S., Schmüdgen, K.: Some selfadjoint \(2\times 2\) operator matrices associated with closed operators. Integral Equ. Oper. Theory 45(4), 475–484 (2003)CrossRefGoogle Scholar
  12. 12.
    Putnam, C.R.: Commutation Properties of Hilbert Space Operators and Related Topics. Springer-Verlag, New York (1967)CrossRefGoogle Scholar
  13. 13.
    Reed, M., Simon, B.: Methods of Modern Mathematical Physics, Vol. 1: Functional Analysis. Academic Press, Cambridge (1972)zbMATHGoogle Scholar
  14. 14.
    Schmüdgen, K.: Unbounded Operator Algebras and Representation Theory. Operator Theory: Advances and Applications, vol. 37. Birkhäuser Verlag, Basel (1990)CrossRefGoogle Scholar
  15. 15.
    Schmüdgen, K.: Unbounded Self-Adjoint Operators on Hilbert Space, vol. 2. Springer, Berlin (2012)CrossRefGoogle Scholar
  16. 16.
    Tretter, Ch.: Spectral Theory of Block Operator Matrices and Applications. Imperial College Press, London (2008)CrossRefGoogle Scholar
  17. 17.
    Weidmann, J.: Linear Operators in Hilbert Spaces. Springer, Berlin (1980)CrossRefGoogle Scholar

Copyright information

© Springer Nature Switzerland AG 2019

Authors and Affiliations

  1. 1.Department of MathematicsUniversity of Oran 1El Menouar OranAlgeria
  2. 2.OranAlgeria

Personalised recommendations