Revista de la
Unión Matemática Argentina
Drazin invertibility of linear operators on quaternionic Banach spaces
El Hassan Benabdi and Mohamed Barraa

Volume 65, no. 2 (2023), pp. 277–284    

Published online: October 31, 2023

https://doi.org/10.33044/revuma.2700

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Abstract

The paper studies the Drazin inverse for right linear operators on a quaternionic Banach space. Let $A$ be a right linear operator on a two-sided quaternionic Banach space. It is shown that if $A$ is Drazin invertible then the Drazin inverse of $A$ is given by $f(A)$, where $f$ is $0$ in an axially symmetric neighborhood of $0$ and $f(q) = q^{-1}$ in an axially symmetric neighborhood of the nonzero spherical spectrum of $A$. Some results analogous to the ones concerning the Drazin inverse of operators on complex Banach spaces are proved in the quaternionic context.

References

  1. F. Colombo, J. Gantner, and D. P. Kimsey, Spectral Theory on the S-Spectrum for Quaternionic Operators, Operator Theory: Advances and Applications 270, Birkhäuser/Springer, Cham, 2018.  DOI  MR  Zbl
  2. M. P. Drazin, Pseudo-inverses in associative rings and semigroups, Amer. Math. Monthly 65 (1958), 506–514.  DOI  MR  Zbl
  3. R. Ghiloni, V. Moretti, and A. Perotti, Continuous slice functional calculus in quaternionic Hilbert spaces, Rev. Math. Phys. 25 no. 4 (2013), 1350006, 83 pp.  DOI  MR  Zbl
  4. I. Gohberg, S. Goldberg, and M. A. Kaashoek, Classes of Linear Operators. Vol. I, Operator Theory: Advances and Applications 49, Birkhäuser Verlag, Basel, 1990.  DOI  MR  Zbl
  5. J. J. Koliha, A generalized Drazin inverse, Glasgow Math. J. 38 no. 3 (1996), 367–381.  DOI  MR  Zbl
  6. B. Muraleetharan and K. Thirulogasanthar, Weyl and Browder S-spectra in a right quaternionic Hilbert space, J. Geom. Phys. 135 (2019), 7–20.  DOI  MR  Zbl