Jacobson’s Lemma in the ring of quaternionic linear operators

In the present paper, we study the Jacobson’s Lemma in the unital ring of all bounded right linear operators ℬ_R(X) acting on a two-sided quaternionic Banach space X. In particular, let A, B ∈ ℬ_R(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.