Local form-subordination condition and Riesz basisness of root systems

We exploit the so called form-local subordination in the analysis of non-symmetric perturbations of unbounded self-adjoint operators with isolated simple positive eigenvalues. If the proper condition relating the size of gaps between the unperturbed eigenvalues and the strength of perturbation, measured by the form-local subordination, is satisfied, the root system of the perturbed operator contains a Riesz basis and usual asymptotic formulas for perturbed eigenvalues and eigenvectors hold. The power of the abstract perturbation results is demonstrated particularly on Schr\"odinger operators with possibly unbounded or singular complex potential perturbations.