Abstract
Let \(\mathcal {A}\) be a Banach algebra and let \(J \subset \mathcal {A}\) be a set that is contained in the set of all quasinilpotent elements of \(\mathcal {A}\). We say that \(a \in \mathcal {A}\) is Drazin invertible relative to J if there exists \(b \in \mathcal {A}\) commuting with a such that \(b=bab\) and \(a-aba \in J\). Given a homomorphism \(T: \mathcal {A}\rightarrow \mathcal {B}\) between Banach algebras, we give necessary and sufficient conditions for Ta to be Drazin invertible relative to J. What is more, several applications to the case of the Calkin homomorphism are indicated. In particular, we give a partial answer to a question related to the structure of polynomially compact operators on Banach spaces.
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The authors would like to thank the referee for his useful comments and observations
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Dijana Mosić and Snežana Č. Živković-Zlatanović are supported by the Ministry of Education, Science and Technological Development, Republic of Serbia, grant no. 451-03-68/2022-14/200124.
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Cvetković, M.D., Mosić, D. & Živković-Zlatanović, S.Č. Drazin Invertibility Relative to Some Subsets of Quasinilpotents and Homomorphism Ranges. Results Math 78, 69 (2023). https://doi.org/10.1007/s00025-023-01848-z
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DOI: https://doi.org/10.1007/s00025-023-01848-z