Common properties of bounded linear operators AC and BA: Spectral theory

Let X, Y be Banach spaces, A:X→Y and B, C:Y→X be bounded linear operators satisfying the operator equation ABA=ACA. Recently, as extensions of Jacobson's lemma, Corach, Duggal and Harte studied common properties of AC−I and BA−I in algebraic viewpoint and also obtained some topological analogue...

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Veröffentlicht in:Mathematische Nachrichten 2014-04, Vol.287 (5-6), p.717-725
Hauptverfasser: Zeng, Qingping, Zhong, Huaijie
Format: Artikel
Sprache:eng
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Zusammenfassung:Let X, Y be Banach spaces, A:X→Y and B, C:Y→X be bounded linear operators satisfying the operator equation ABA=ACA. Recently, as extensions of Jacobson's lemma, Corach, Duggal and Harte studied common properties of AC−I and BA−I in algebraic viewpoint and also obtained some topological analogues. In this note, we continue to investigate common properties of AC and BA from the viewpoint of spectral theory. In particular, we give an affirmative answer to one question posed by Corach et al. by proving that AC−I has closed range if and only if BA−I has closed range.
ISSN:0025-584X
1522-2616
DOI:10.1002/mana.201300123