Moroccan Journal of Pure and Applied Analysis (Sep 2021)

Jacobson’s Lemma in the ring of quaternionic linear operators

  • Benabdi El Hassan,
  • Barraa Mohamed

DOI
https://doi.org/10.2478/mjpaa-2021-0031
Journal volume & issue
Vol. 7, no. 3
pp. 461 – 469

Abstract

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In the present paper, we study the Jacobson’s Lemma in the unital ring of all bounded right linear operators ℬR(X) acting on a two-sided quaternionic Banach space X. In particular, let A, B ∈ ℬR(X) and let q ∈ ℍ \ {0}, we prove that w(AB) \ {0} = w(BA) \ {0} where w belongs to the spherical spectrum, the spherical approximate point spectrum, the right spherical spectrum, the left spherical spectrum, the spherical point spectrum, the spherical residual spectrum and the spherical continuous spectrum. We also prove that the range of (AB)2 − 2Re(q)AB + |q|2I is closed if and only if (BA)2 − 2Re(q)BA + |q|2I has closed range. Finally, we show that (AB)2 − 2Re(q)AB + |q|2I is Drazin invertible if and only if (BA)2 − 2Re(q)BA + |q|2I is.

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