Abstract
We study the stability of certain spectra under some algebraic conditions weaker than the commutativity and we generalize many known commutative perturbation results. In particular, in a complex unital Banach algebra \(\mathcal {A},\) we prove that if \(x \in \hbox {comm}_{w}(a)\) and a is nilpotent, then \(\sigma (x)=\sigma (x+a).\) Among other things, we prove that if \(x \in \hbox {comm}(xy)\cup \hbox {comm}(yx)\) or \(y\in \hbox {comm}(yx)\) for all \(x,y \in \mathcal {A},\) then \(\mathcal {A}\) is commutative. When \(\mathcal {A}=L(X)\) is the unital complex Banach algebra of bounded linear operators acting on the complex Banach space X, then \(\sigma _{p}(T){{\setminus }}\{0\}\subset \sigma _{p}(T+N){\setminus }\{0\},\) where N is nilpotent and \(T\in \hbox {comm}(NT).\)
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The authors would like to thank the referees and the Editor-in-chief for important comments and corrections that have improved this article.
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Communicated by Dragana Cvetkovic Ilic.
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Aznay, Z., Ouahab, A. & Zariouh, H. Generalization of some commutative perturbation results. Ann. Funct. Anal. 14, 6 (2023). https://doi.org/10.1007/s43034-022-00231-3
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DOI: https://doi.org/10.1007/s43034-022-00231-3