A Generalization of Strongly Preserver Problems of Drazin Invertibility

Abstract

Let ϕ be an additive map between unital complex Banach algebras such that ϕ(1) is invertible. We show that ϕ satisfies ϕ(aD)ϕ(b)D = ϕ(a)Dϕ(bD) for every Drazin invertible elements a, b if and only if ϕ(1)− 1ϕ 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.

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Correspondence to Khalid Souilah.

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Oudghiri, M., Souilah, K. A Generalization of Strongly Preserver Problems of Drazin Invertibility. Acta Math Vietnam 43, 575–583 (2018). https://doi.org/10.1007/s40306-018-0251-6

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Keywords

  • Linear preserver problems
  • Jordan homomorphism
  • Drazin invertibility

Mathematics Subject Classification (2010)

  • 47B48
  • 47B49
  • 47L99
  • 46H05