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 operator T ( ε ) : = T 0 + ε T 1 + ε 2 T 2 + ⋯ + ε k T k + ⋯ , forms an unconditional basis with parentheses in a separable Hilbert space X; where ε ∈ C , T 0 is a closed densely defined linear operator on X with domain D ( T 0 ) , having compact resolvent, while T 1 , T 2 , … are linear operators on X, with the same domain D ⊃ D ( T 0 ) , satisfying a specific growing inequality. An application to a problem of radiation of a vibrating structure in a light fluid is presented.