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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.

The basis property of eigenfunctions in the problem of a nonhomogeneous damped string

Rzepnicki Ł.

Opuscula Mathematica

2017

Vol. 37, no. 1

141--165

The equation which describes the small vibrations of a nonhomogeneous damped string can be rewritten as an abstract Cauchy problem for the densely defined closed operator iA. We prove that the set of root vectors of the operator A forms a basis of subspaces in a certain Hilbert space H. Furthermore, we give the rate of convergence for the decomposition with respect to this basis. In the second main result we show that with additional assumptions the set of root vectors of the operator A is a Riesz basis for H.