Perturbations of normal operators and applications (2023)
We present some new results for normal operators. We study the effect of relatively bounded perturbations on the spectrum of normal operators and present stability results for spectral gaps considering the spectrum of the unperturbated operator is close to the real axis or it is contained in a sector symmetric to the real axis. we do not assume that the perturbation is symmetric or normal. Moreover, we established stability results for spectral gaps, essential spectrum gaps, algebraic multiplicities and estimates for the resolvent. If the perturbation is even p-subordinate to the normal operator, then we obtain stronger results for the localisation of the spectrum. Finally, we apply some of our results to Schrödinger operators. First, we consider a Laplacian defined on quantum star graph with non-selfadjoint boundary conditions. Next, we consider a Laplacian with quasiperiodic boundary conditions.