Advances in Operator Theory

On the numerical radius of a quaternionic normal operator

Golla Ramesh

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Abstract

We prove that for a right linear bounded normal operator on a quaternionic Hilbert space (quaternionic bounded normal operator) the norm and the numerical radius are equal. As a consequence of this result we give a new proof of the known fact that a non zero quaternionic compact normal operator has a non zero right eigenvalue. Using this we give a new proof of the spectral theorem for quaternionic compact normal operators. Finally, we show that every quaternionic compact operator is norm attaining and prove the Lindenstrauss theorem on norm attaining operators, namely, the set of all norm attaining quaternionic operators is norm dense in the space of all bounded quaternionic operators defined between two quaternionic Hilbert spaces.

Article information

Source
Adv. Oper. Theory, Volume 2, Number 1 (2017), 78-86.

Dates
Received: 22 November 2016
Accepted: 26 January 2017
First available in Project Euclid: 4 December 2017

Permanent link to this document
https://projecteuclid.org/euclid.aot/1512431515

Digital Object Identifier
doi:10.22034/aot.1611-1060

Mathematical Reviews number (MathSciNet)
MR3730356

Zentralblatt MATH identifier
1367.47007

Subjects
Primary: 47S10: Operator theory over fields other than $R$, $C$ or the quaternions; non- Archimedean operator theory
Secondary: 43B15 35P05: General topics in linear spectral theory

Keywords
quaternionic Hilbert space normal operator compact operator right eigenvalue norm attaining operator Lindenstrauss theorem

Citation

Ramesh, Golla. On the numerical radius of a quaternionic normal operator. Adv. Oper. Theory 2 (2017), no. 1, 78--86. doi:10.22034/aot.1611-1060. https://projecteuclid.org/euclid.aot/1512431515


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References

  • D. R. Farenick and B. A. F. Pidkowich, The spectral theorem in quaternions, Linear Algebra Appl. 371 (2003), 75–102.
  • R. Ghiloni, V. Moretti, and A. Perotti, Continuous slice functional calculus in quaternionic Hilbert spaces, Rev. Math. Phys. 25 (2013), no. 4, 1350006, 83 pp.
  • Y. H. Au-Yeung, On the convexity of numerical range in quaternionic Hilbert spaces, Linear and Multilinear Algebra 16 (1984), no. 1-4, 93–100.
  • R. Ghiloni, V. Moretti, and A. Perotti. Spectral properties of compact normal quaternionic operators, Bernstein, Swanhild (ed.) et al., Hypercomplex analysis: new perspectives and applications. Selected papers presented at the session on Clifford and quaternionic analysis at the 9th congress of the International Society for Analysis, its Applications, and Computation (ISAAC), Krakow, Poland, August 5–9, 2013. New York, NY: Birkhäuser/Springer. Trends in Mathematics, 133–143 (2014).
  • M. Fashandi, Compact operators on quaternionic Hilbert spaces, Facta Univ. Ser. Math. Inform. 28 (2013), no. 3, 249–256.
  • P. Enflo, J. Kover, and L. Smithies, Denseness for norm attaining operator-valued functions, Linear Algebra Appl. 338 (2001), 139–144.