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On a Schauder basis related to the eigenvectors of a family of non-selfadjoint analytic operators and applications

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Abstract

In the present paper, we deal with the perturbed operator

$$\begin{aligned} T(\varepsilon ):=T_0+\varepsilon T _1+\varepsilon ^2T_2+\cdots +\varepsilon ^kT_k+\cdots , \end{aligned}$$

where \(\varepsilon \in \mathbb C ,\,T_0\) is a closed densely defined linear operator on a separable Banach space \(X\) with domain \(\mathcal D (T_0),\) while \(T_1, T_2, \ldots \) are linear operators on \(X\) with 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 assuring the invariance of the closure of the perturbed operator \(T(\varepsilon )\) which enables us to study the changed spectrum. Moreover, we prove that the system formed by some eigenvectors of \(T(\varepsilon )\) which are analytic on \(\varepsilon ,\) forms a Schauder basis in \(X.\) After that, we apply the obtained results to a nonself-adjoint problem describing the radiation of a vibrating structure in a light fluid and to a nonself-adjoint Gribov operator in Bargmann space.

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Authors and Affiliations

  1. Département de Mathématiques, Université de Sfax, Faculté des sciences de Sfax, Route de soukra Km 3.5, B.P. 1171, 3000 , Sfax, Tunisie

    Ines Feki, Aref Jeribi & Ridha Sfaxi

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  1. Ines Feki
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  2. Aref Jeribi
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  3. Ridha Sfaxi
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Correspondence to Aref Jeribi.

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Feki, I., Jeribi, A. & Sfaxi, R. On a Schauder basis related to the eigenvectors of a family of non-selfadjoint analytic operators and applications. Anal.Math.Phys. 3, 311–331 (2013). https://doi.org/10.1007/s13324-013-0060-y

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  • DOI: https://doi.org/10.1007/s13324-013-0060-y

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