An An embedding theorem and spectral equality for semigroups involving demicompactness classes
Resumen
Let $(S(t))_{t\geq0}$ and $(T(t))_{t\geq0}$ denote the strongly continuous semigroups of operators in a Banach space $X$. In this paper, we give a sufficient condition guaranteeing that $(S(t))_{t\geq0}$ can be embedded in a $C_{0}$-group on $X$. Moreover, we characterize the demicompactness of $I-(S(t)-T(t))$ for $t>0$. Our theoretical results will be illustrated by investigating the spectral equality for uniformly continuous semigroups for an upper semi-Fredholm spectrum.
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Jeribi A., Krichen B., Salhi M.: \emph{Characterization of Relatively Demicompact Operators by Means of Measures of Noncompactness}. J. Korean Math. Soc., \textbf{55}, 877-895 (2018)
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Latrach K., Dehici A.: \emph{Remarks on embeddable semigroups in groups and a generalization of some Cuthbert's results}. Int. J. Math. Math. Sci., \textbf{22}, 1421-1431 (2003)
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Pazy A.: \emph{Semigroups of linear operators and applications to partial differential equations}. Applied Mathematical Sciences, Springer-Verlag, New York, \textbf{44}, (1983)
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Petryshyn W. V.: \emph{Construction of fixed points of demicompact mappings in Hilbert space}. J. Math. Anal. Appl., \textbf{14},
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Williams V.: \emph{Closed Fredholm and Semi-Fredholm Operators, Essential Spectra and Perturbations}. J. Functional Anal., \textbf{20}, 1-25 (1975)
Benkhaled H., Elleuch A., Jeribi A.: \emph{Demicompactness Results for Strongly Continuous Semigroups, Generators and Resolvents}. Mediterr. J. Math., \textbf{15}(2), (2018)
\bibitem{B.E.J.2}
Benkhaled H., Elleuch A., Jeribi A.: \emph{Demicompactness Properties for Uniformly Continuous Cosine Families}. To appear in Mediterr. J. Math., (2022)
\bibitem{E.N}
Engel K. J., Nagel R.: \emph{One-parameter Semigroups for Linear Evolution Equations}. pringer-Verlag, (2000)
\bibitem{H.P}
Hille E., Phillips R. S.: \emph{Functional Analysis and Semi-Groups}. American Mathematical Society Colloquium Publications, American Mathematical Society, Rhode Island, \textbf{31}, (1957)
\bibitem{J}
Jeribi A.: \emph{Spectral theory and applications of linear operators and block operator matrices}, Springer-Verlag, New-York, (2015)
\bibitem{J1}
Jeribi A., Krichen B., Salhi M.: \emph{Characterization of Relatively Demicompact Operators by Means of Measures of Noncompactness}. J. Korean Math. Soc., \textbf{55}, 877-895 (2018)
\bibitem{K.K.1}
Kuratowski K.: \emph{Sur les espaces complets}. Fund. Math., \textbf{15}, 301-309 (1930)
\bibitem{K.K.2}
Kuratowski K.: \emph{Topology}. Hafner, New York, (1966)
\bibitem{L}
Latrach K., Dehici A.: \emph{Remarks on embeddable semigroups in groups and a generalization of some Cuthbert's results}. Int. J. Math. Math. Sci., \textbf{22}, 1421-1431 (2003)
\bibitem{A.P}
Pazy A.: \emph{Semigroups of linear operators and applications to partial differential equations}. Applied Mathematical Sciences, Springer-Verlag, New York, \textbf{44}, (1983)
\bibitem{P}
Petryshyn W. V.: \emph{Construction of fixed points of demicompact mappings in Hilbert space}. J. Math. Anal. Appl., \textbf{14},
276-284 (1966)
\bibitem{W}
Williams V.: \emph{Closed Fredholm and Semi-Fredholm Operators, Essential Spectra and Perturbations}. J. Functional Anal., \textbf{20}, 1-25 (1975)
Publicado
2025-10-31
Número
Sección
Research Articles
Derechos de autor 2025 Boletim da Sociedade Paranaense de Matemática
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