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Explores new g-Hirano inverses, revealing conditions for invertibility in Banach algebras, indicating their properties.
In aBanach algebra, we introduce a new type of generalized inverse called g?-Hirano inverse. Firstly, several existence criteria and the equivalent definition of this inverse are investigated. Then, we dis-cuss the relationship between the g?-Hirano invertibility of a, b and that of the sum a + b under some weaker conditions. Finally, as applications to the previous additive results, some equivalent characterizations for the g?-Hirano invertibility of the anti-triangular matrix over Banach algebras are obtained.
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Zou et al. (2025) studied this question.
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