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1 Journal of Mathematics and Applications

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On the basis property of root vectors related to a non-self-adjoint analytic operator and applications

Ellouz Hanen, Feki Ines, Jeribi Aref

Journal of Mathematics and Applications

2022

Vol. 45

59--86

In the present paper, based on a separation condition on the spectrum of a self-adjoint operator T0 on a separable Hilbert space H, we prove that the system of root vectors of the perturbed operator T (ε) given by T (ε) := T0 + εT1 + ε2T2 + . . . + εkTk + . . . is complete and forms a basis with parentheses in H, for small enough |ε|. Here ε ∈ C and T1, T2, . . . are linear operators on H having the same domain D ⊃ D(T0) and satisfying a specific growing inequality. The obtained results are of importance for applications to a non-self-adjoint Gribov operator in Bargmann space and to a non-self-adjoint problem deduced from a perturbation method for sound radiation.