Börje Nilsson: Basis properties for sets of eigenfunctions to non-self-adjoint operators

Basis properties for sets of eigenfunctions to self-adjoint operators in a separable Hilbert space can be proved based on a well-developed functional calculus. The situation is much more complicated for the non-self-adjoint case. We present a review of classical basis results for non-self adjoint operators, including general basis theory, perturbation of orthogonal bases, and almost self-adjoint as well as dissipative operators. The results are discussed in relation to a simple example with an ordinary differential operator and in relation to an application on electromagnetic waveguide theory used for the mathematical modelling of high voltage power cables with dissipation.