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On a Riesz basis of exponentials related to a family of analytic operators and application

  • Hanen Ellouz
  • Ines Feki
  • Aref JeribiEmail author
Article
  • 64 Downloads

Abstract

In this paper, we are interested by the perturbed operator
$$\begin{aligned} T(\varepsilon ):=T_0+\varepsilon T_1 +\varepsilon ^2T_2+\cdots +\varepsilon ^k T_k+\cdots \end{aligned}$$
where \(\varepsilon \in \mathbb {C}\), \(T_0\) is a closed densely defined linear operator on a separable Hilbert space \(\mathcal{H}\) with domain \(\mathcal{D}(T_0)\) having isolated eigenvalues with multiplicity one whereas \(T_1, T_2,\ldots \) are linear operators on \(\mathcal{H}\) having the same domain \(\mathcal{D}\supset \mathcal{D}(T_0)\) and satisfying a specific growing inequality. The basic idea here is to investigate under sufficient conditions the existence of Riesz bases of exponentials, where the exponents corresponding as a sequence of eigenvalues of \(T(\varepsilon )\), can be developed as entire series of \(\varepsilon \). An application to a nonself-adjoint problem describing the radiation of a vibrating structure in a light fluid is presented.

Keywords

Eigenvalues Elastic membrane Families of exponentials Isolated point Riesz bases 

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Copyright information

© Springer Nature Switzerland AG 2018

Authors and Affiliations

  1. 1.Département de MathématiquesUniversité de Sfax Faculté des sciences de SfaxSfaxTunisia

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