Advertisement

Complex Analysis and Operator Theory

, Volume 13, Issue 1, pp 85–113 | Cite as

Deficiency Indices of Some Classes of Unbounded \(\mathbb H\)-Operators

  • B. Muraleetharan
  • K. ThirulogasantharEmail author
Article

Abstract

In this paper we define the deficiency indices of a closed symmetric right \(\mathbb H\)-linear operator and formulate a general theory of deficiency indices in a right quaternionic Hilbert space. This study provides a necessary and sufficient condition in terms of deficiency indices and in terms of S-spectrum, parallel to their complex counterparts, for a symmetric right \(\mathbb H\)-linear operators to be self-adjoint.

Keywords

Quaternions Quaternionic Hilbert spaces Symmetric operator Deficiency index S-spectrum 

Mathematics Subject Classification

Primary 47B32 47S10 

References

  1. 1.
    Adler, S.L.: Quaternionic Quantum Mechanics and Quantum Fields. Oxford University Press, New York (1995)zbMATHGoogle Scholar
  2. 2.
    Ali, S.T., Antoine, J.-P., Gazeau, J.-P.: Coherent States, Wavelets and Their Generalizations. Springer, New York (2000)CrossRefzbMATHGoogle Scholar
  3. 3.
    Alpay, D., Colombo, F., Kimsey D.P.: The spectral theorem for quaternionic unbounded normal operators based on the S-spectrum. J. Math. Phys. 57, 023503 (2016)Google Scholar
  4. 4.
    Colombo, F., Sabadini, I.: On some properties of the quaternionic functional calculus. J. Geom. Anal. 19, 601–627 (2009)MathSciNetCrossRefzbMATHGoogle Scholar
  5. 5.
    Colombo, F., Sabadini, I.: On the formulations of the quaternionic functional calculus. J. Geom. Phys. 60, 1490–1508 (2010)MathSciNetCrossRefzbMATHGoogle Scholar
  6. 6.
    Gazeau, J.-P.: Coherent States in Quantum Physics. Wiley, Berlin (2009)CrossRefGoogle Scholar
  7. 7.
    Ghiloni, R., Moretti, W., Perotti, A.: Continuous slice functional calculus in quaternionic Hilbert spaces. Rev. Math. Phys. 25, 1350006 (2013)MathSciNetCrossRefzbMATHGoogle Scholar
  8. 8.
    Schmdgen, K.: Unbounded Self-Adjoint Operators on Hilbert Space. Springer, Netherlands (2012)CrossRefGoogle Scholar
  9. 9.
    Vasilescu, F.-H.: Quaternionic Cayley transform revisited. J. Math. Anal. Appl. 409, 790–807 (2014)MathSciNetCrossRefzbMATHGoogle Scholar
  10. 10.
    Viswanath, K.: Normal operators on quaternionic Hilbert spaces. Trans. Am. Math. Soc. 162, 337–350 (1971)zbMATHGoogle Scholar
  11. 11.
    Zhang, F.: Quaternions and matrices of quaternions. Linear Algebra Appl. 251, 21–57 (1997)MathSciNetCrossRefzbMATHGoogle Scholar

Copyright information

© Springer International Publishing AG 2017

Authors and Affiliations

  1. 1.Department of Mathematics and StatisticsUniversity of JaffnaThirunelveli, JaffnaSri Lanka
  2. 2.Department of Computer Science and Software EngineeringConcordia UniversityMontrealCanada

Personalised recommendations