On an unconditional basis of generalized eigenvectors of an analytic operator and application to a problem of radiation of a vibrating structure in a light fluid

In this paper, we prove that the system of generalized eigenvectors of the perturbed operatorT(ε):=T0+εT1+ε2T2+⋯+εkTk+⋯, forms an unconditional basis with parentheses in a separable Hilbert space X; where ε∈C, T0 is a closed densely defined linear operator on X with domain D(T0), having compact resolvent, while T1,T2,… are linear operators on X, with the same domain D⊃D(T0), satisfying a specific growing inequality. An application to a problem of radiation of a vibrating structure in a light fluid is presented.