<abstract xmlns="http://www.w3.org/1999/xhtml">

In the present paper, we study the Jacobson’s Lemma in the unital ring of all bounded right linear operators ℬ<italic>R</italic>(<italic>X</italic>) acting on a two-sided quaternionic Banach space <italic>X</italic>. In particular, let <italic>A</italic>, <italic>B</italic> ∈ ℬ<italic>R</italic>(<italic>X</italic>) and let <italic>q</italic> ∈ ℍ \ {0}, we prove that <italic>w</italic>(<italic>AB</italic>) \ {0} = <italic>w</italic>(<italic>BA</italic>) \ {0} where <italic>w</italic> 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 (<italic>AB</italic>)2 − 2Re(<italic>q</italic>)<italic>AB</italic> + |<italic>q</italic>|2<italic>I</italic> is closed if and only if (<italic>BA</italic>)2 − 2Re(<italic>q</italic>)<italic>BA</italic> + |<italic>q</italic>|2<italic>I</italic> has closed range. Finally, we show that (<italic>AB</italic>)2 − 2Re(<italic>q</italic>)<italic>AB</italic> + |<italic>q</italic>|2<italic>I</italic> is Drazin invertible if and only if (<italic>BA</italic>)2 − 2Re(<italic>q</italic>)<italic>BA</italic> + |<italic>q</italic>|2<italic>I</italic> is.
</abstract>

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.

Jacobson’s Lemma in the ring of quaternionic linear operators

Department of Mathematics, Faculty of Sciences Semlalia

Moroccan Journal of Pure and Applied Analysis

This work is licensed under the Creative Commons Attribution-NonCommercial-NoDerivatives 4.0 International License.

{"article-title":"Jacobson\u2019s 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 ℬR(X) acting on a two-sided quaternionic Banach...

Jacobson’s Lemma in the ring of quaternionic linear operators

Published Online: Apr 30, 2021

Page range: 461 - 469

Received: Jun 18, 2020

Accepted: Apr 13, 2021

DOI: https://doi.org/10.2478/mjpaa-2021-0031

Keywords
Quaternionic Banach space, Spherical spectrum, Quaternionic Banach algebra

© 2021 El Hassan Benabdi et al., published by Sciendo

This work is licensed under the Creative Commons Attribution-NonCommercial-NoDerivatives 4.0 International License.

Jacobson’s Lemma in the ring of quaternionic linear operators

Published Online: Apr 30, 2021

Page range: 461 - 469

Received: Jun 18, 2020

Accepted: Apr 13, 2021

DOI: https://doi.org/10.2478/mjpaa-2021-0031

KeywordsQuaternionic Banach space, Spherical spectrum, Quaternionic Banach algebra

© 2021 El Hassan Benabdi et al., published by Sciendo

This work is licensed under the Creative Commons Attribution-NonCommercial-NoDerivatives 4.0 International License.

Keywords
Quaternionic Banach space, Spherical spectrum, Quaternionic Banach algebra