On the Weak Topology of Quaternionic Hilbert Spaces

  • M. FashandiEmail author


In this paper, we study weak topology on right quaternionic Hilbert spaces. We show that on right quaternionic Hilbert spaces weak compactness and weak sequential compactness coincide which could be viewed as a generalization of Eberlein–Šmulian theorem. Also, we apply this observation on study compact operators on right quaternionic Hilbert spaces.


Quaternionic Hilbert spaces Weak topology Compact operators 

Mathematics Subject Classification

46E22 47B07 



The author highly appreciates the reviewers’ insightful and valuable comments. Also, the author thanks Professor Jan Lang from the Ohio State University, for his helpful suggestions on the initial version of this manuscript. This research was supported by a Grant from Ferdowsi University of Mashhad: No. 2/47516.


  1. 1.
    Adler, S.L.: Quaternionic Quantum Mechanics and Quantum Fields. Oxford University Press, New York (1995)zbMATHGoogle Scholar
  2. 2.
    Alpay, D., Colombo, F., Sabadini, I.: Slice Hyperholomorphic Schur Analysis. Birkhäuser, Basel (2016)CrossRefGoogle Scholar
  3. 3.
    Alpay, D., Colombo, F., Kimsey, D.: The spectral theorem for quaternionic unbounded normal operators based on the S-spectrum. J. Math. Phys. 57, 023503 (2016)ADSMathSciNetCrossRefGoogle Scholar
  4. 4.
    Alpay, D., Colombo, F., Sabadini, I.: On a class of quaternionic positive definite functions and their derivatives. J. Math. Phys. 58, 033501 (2017)ADSMathSciNetCrossRefGoogle Scholar
  5. 5.
    Alpay, D., Shapiro, M.: Reproducing kernel quaternionic pontryagin spaces. Integr. Equ. Oper. Theory 50, 431–476 (2004)MathSciNetCrossRefGoogle Scholar
  6. 6.
    Bauschke, H.H., Combettes, P.L.: Convex Analysis and Monotone Operator Theory in Hilbert Spaces. Springer, Berlin (2011)CrossRefGoogle Scholar
  7. 7.
    Cheng, D., Kou, K.I.: Novel sampling formulas associated with quaternionic prolate spheroidal wave functions. Adv. Appl. Clifford Algebras 27, 2961–2983 (2017)MathSciNetCrossRefGoogle Scholar
  8. 8.
    Colombo, F., Sabadini, I., Struppa, D.C.: Noncommutative functional calculus. In: Theory and Applications of Slice Hyperholomorphic Functions, Volume 289 of Progress in Mathematics. Birkhäuser/Springer Basel AG, Basel (2011)Google Scholar
  9. 9.
    Colombo, F., Sabadini, I.: On some notions of spectra for quaternionic operators and for n-tuples of operators. C. R. Math. Acad. Sci. Paris 350, 399402 (2012)MathSciNetCrossRefGoogle Scholar
  10. 10.
    Colombo, F., Gantner, J., Janssen, T.: Schatten class and Berezin transform of quaternionic linear operators. Math. Methods Appl. Sci. 39, 5582–5606 (2016)ADSMathSciNetCrossRefGoogle Scholar
  11. 11.
    Conway, J.B.: A Course in Functional Analysis. Springer, Berlin (1985)CrossRefGoogle Scholar
  12. 12.
    Edmunds, D.E., Evans, W.D.: Representations of linear operators between Banach spaces. In: Operator Theory: Advances and Applications, vol. 238. Birkhäuser/Springer, Basel (2013)Google Scholar
  13. 13.
    Fashandi, M.: Compact operators on quaternionic Hilbert spaces. Facta Univ. Ser. Math. Inform. 28, 249–256 (2013)MathSciNetzbMATHGoogle Scholar
  14. 14.
    Fashandi, M.: Some properties of bounded linear operators on quaternionic Hilbert spaces. Kochi J. Math. 9, 127–135 (2014)MathSciNetzbMATHGoogle Scholar
  15. 15.
    Ghiloni, R., Moretti, V., Perotti, A.: Continuous slice functional calculus in quaternionic Hilbert spaces. Rev. Math. Phys. 25, 1350006 (2013)MathSciNetCrossRefGoogle Scholar
  16. 16.
    Ghiloni, R., Moretti, V., Perotti, A.: Spectral properties of compact normal quaternionic operators. In: Bernstein, S., Kähler, U., Sabadini, I., Sommen, F. (eds.) Hypercomplex Analysis: New Perspectives and Applications, Trends in Mathematics, pp. 133–143. Birkhäuser, Basel (2014)Google Scholar
  17. 17.
    Komornik, V.: Lectures on Functional Analysis and the Lebesgue Integral. Universitext: Springer, Berlin (2016)CrossRefGoogle Scholar
  18. 18.
    Ramesh, G.: On the numerical radius of a quaternionic normal operator. Adv. Oper. Theory 2, 78–86 (2017)MathSciNetzbMATHGoogle Scholar
  19. 19.
    Teichmuller, O.: Operatoren im Wachsschen Raum. J. Reine Angew. Math. 174, 73–124 (1936)MathSciNetzbMATHGoogle Scholar
  20. 20.
    Viswanath, K.: Contributions to Linear Quaternionic Analysis. Ph.D. Thesis, Indian Statistical Institute, Calcutta (1968)Google Scholar

Copyright information

© Springer Nature Switzerland AG 2018

Authors and Affiliations

  1. 1.Faculty of Mathematical SciencesFerdowsi University of MashhadMashhadIran

Personalised recommendations