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Local spectral theory for \(\alpha \)-times integrated semigroups

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Abstract

In this work, we study the local spectral theory for \(\alpha \)-times integrated semigroups \(\{T(t)\}_{t\ge 0}\) on a Banach space X for \(\alpha \ge 0\). More precisely, we show that \(\{T(t)\}_{t\ge 0}\) has the SVEP (single valued extension property) if and only if their generator A also has the SVEP. Furthermore, we give some local spectral inclusions between \(\{T(t)\}_{t\ge 0}\) and their generator A. Also, we establish the spectral inclusion of \(\{T(t)\}_{t\ge 0}\) for left and right spectra. Finally, we investigate the transmission of some properties from a \(C_0\)-semigroup to their infinitesimal generator.

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Acknowledgements

The authors are thankful to the referee for his valuable comments and suggestions.

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  1. Laboratory of Mathematical Sciences and Applications (LaSMA), Sidi Mohamed Ben Abdellah University, Faculty of Sciences Dhar Al Mahraz, Fez, Morocco

    A. Tajmouati, F. Barki & M. A. O. M. Baba

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  1. A. Tajmouati
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  2. F. Barki
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  3. M. A. O. M. Baba
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Correspondence to A. Tajmouati.

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Tajmouati, A., Barki, F. & Baba, M.A.O.M. Local spectral theory for \(\alpha \)-times integrated semigroups. Rend. Circ. Mat. Palermo, II. Ser 71, 21–37 (2022). https://doi.org/10.1007/s12215-021-00630-w

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  • DOI: https://doi.org/10.1007/s12215-021-00630-w

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