On a Riesz basis of exponentials related to the eigenvalues of an analytic operator and application to a non-selfadjoint problem deduced from a perturbation method for sound radiation

Abstract

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 ɛ∈C, T0 is a closed densely defined linear operator on a separable Hilbert space H with domain D(T0) having isolated eigenvalues with multiplicity one, while T1, T2, … are linear operators on H having the same domain 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.