Frame of Exponentials Related to Analytic Families Operators and Application to a Non-self Adjoint Problem of Radiation of a Vibrating Structure in a Light Fluid

In the present paper, we investigate under sufficient conditions the existence of frames of exponential families, where the exponents coincide with the eigenvalues of the perturbed operator T(epsilon) := T-0 + epsilon T-1 + epsilon T-2(2) + ... + epsilon T-k(k) + ..., epsilon is an element of C. Here T0 is a closed densely defined linear operator on a separable Hilbert space H with domain D(T0) having isolated eigenvalues with multiplicity one and T1,T2,... are linear operators on H having the same domain DD(T0) and satisfying a specific growing inequality. The obtained results are applied to a non-self adjoint problem deduced from a perturbation method for sound radiation.