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On a class of Drazin invertible operators for which \(\left( S^{*}\right) ^{2}\left( S^{D}\right) ^{2}=\left( S^{*}S^{D}\right) ^{2}\)

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Abstract

In this paper we introduce and analyze a new class of operators for which \( \big ( S^{*}\big ) ^{2}\big ( S^{D}\big )^{2}=\big ( S^{*}S^{D}\big )^{2}\) for a bounded linear operator S acting on a complex Hilbert space \(\mathcal {H}\) using the Drazin inverse \(S^D\) of S. After establishing the basic properties of such operators. We show some results related to this class on a Hilbert space. In addition, we characterize the direct sum and the tensor product of these operators. At the end of this paper we generalize the Kaplansky theorem’s to this class.

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Acknowledgements

The authors are grateful to the referees for their comments and suggestions.

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Author notes

  1. Messaoud Guesba and Sid Ahmed Ould Ahmed Mahmoud contributed their efforts jointly to this manuscript.

Authors and Affiliations

  1. Mathematics Department, Faculty of Exact Sciences, Echahid Hamma Lakhdar University, 39000, El Oued, Algeria

    Messaoud Guesba

  2. Mathematics Department, College of Science, Jouf University, Sakaka, P.O.Box 2014, Saudi Arabia

    Sid Ahmed Ould Ahmed Mahmoud

Authors
  1. Messaoud Guesba
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  2. Sid Ahmed Ould Ahmed Mahmoud
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Correspondence to Messaoud Guesba.

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Guesba, M., Mahmoud, S.A.O.A. On a class of Drazin invertible operators for which \(\left( S^{*}\right) ^{2}\left( S^{D}\right) ^{2}=\left( S^{*}S^{D}\right) ^{2}\). J Anal 32, 1093–1109 (2024). https://doi.org/10.1007/s41478-023-00676-2

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  • DOI: https://doi.org/10.1007/s41478-023-00676-2

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