In the present paper, we prove that the family of exponentials associated to the eigenvalues of the perturbed operator T(ɛ) ≔ T0 + ɛT1 + ɛ2T2 + … + ɛkTk + … forms a Riesz basis in L2(0, T), T > 0, where \documentclass[12pt]{minimal}\begin{document}$\varepsilon \in \mathbb {C}$\end{document}ɛ∈C, T0 is a closed densely defined linear operator on a separable Hilbert space \documentclass[12pt]{minimal}\begin{document}${\cal H}$\end{document}H with domain \documentclass[12pt]{minimal}\begin{document}${\cal D}(T_0)$\end{document}D(T0) having isolated eigenvalues with multiplicity one, while T1, T2, … are linear operators on \documentclass[12pt]{minimal}\begin{document}${\cal H}$\end{document}H having the same domain \documentclass[12pt]{minimal}\begin{document}${\cal D}\supset {\cal D}(T_0)$\end{document}D⊃D(T0) and satisfying a specific growing inequality. After that, we generalize this result using a H-Lipschitz function. As application, we consider a non-selfadjoint problem deduced from a perturbation method for sound radiation.
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