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Some spectral results for demicompact operators and their restrictions with an application to transport equations

Abstract

In this paper, we investigate the \(S_{0_n}\)-demicompactness of the restriction \(T_n\) of a bounded or unbounded linear operator T to \(\mathcal {R}(T^n)\), where \(S_{0_n}\) is the restriction of a given bounded linear operator \(S_0\) to \(\mathcal {R}(T^n)\). The results are formulated in terms of a condition of primality and the closedness of certain ranges. Moreover, we set forward some results on upper semi-Fredholm operators involving weak \(S_0\)-demicompactness class. In particular, we give a new characterization of the \(S_0\)-essential radius and localizations results of some \(S_0\)-essential spectra of T. An example of operator equations arising in transport theory is developed.

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Acknowledgements

The authors wish to thank Pr. M. Berkani for helpful discussions and beneficial comments that motivated many of the results of Sect. 3.

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Correspondence to Bilel Krichen.

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Krichen, B., Trabelsi, B. Some spectral results for demicompact operators and their restrictions with an application to transport equations. Ricerche mat (2022). https://doi.org/10.1007/s11587-022-00688-3

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  • DOI: https://doi.org/10.1007/s11587-022-00688-3

Keywords

  • Weakly \(S_0\)-demicompact operator
  • Quasi Fredholm operators
  • Measure of noncompactness
  • S-essential radius

Mathematics Subject Classification

  • 47A53
  • 47A55