A Generalization of Strongly Preserver Problems of Drazin Invertibility

Let ϕ be an additive map between unital complex Banach algebras such that ϕ (1) is invertible. We show that ϕ satisfies ϕ ( a D ) ϕ ( b ) D = ϕ ( a ) D ϕ ( b D ) for every Drazin invertible elements a , b if and only if ϕ (1) − 1 ϕ is a Jordan homomorphism and ϕ (1) commutes with the range of ϕ . A...

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Bibliographic Details
Published in:Acta mathematica vietnamica 2018-09, Vol.43 (3), p.575-583
Main Authors: Oudghiri, Mourad, Souilah, Khalid
Format: Article
Language:English
Subjects:
Banach spaces
Linear operators
Mathematics
Mathematics and Statistics
Operators (mathematics)
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Summary:Let ϕ be an additive map between unital complex Banach algebras such that ϕ (1) is invertible. We show that ϕ satisfies ϕ ( a D ) ϕ ( b ) D = ϕ ( a ) D ϕ ( b D ) for every Drazin invertible elements a , b if and only if ϕ (1) − 1 ϕ is a Jordan homomorphism and ϕ (1) commutes with the range of ϕ . A similar result is established for group invertible elements, and more explicit forms of such maps are given in the context of the algebra of all bounded linear operators on a complex Banach space.
ISSN:0251-4184
2315-4144
DOI:10.1007/s40306-018-0251-6