Perturbations of positive operators with discrete spectrum. Analytical aspects

We study spectral properties of perturbed operators $A=T+B$, where $B$ is a non-symmetric operator subordinated to a self-adjoint, normal or positive operator $T$. We consider different types of perturbatios: completely subordinated, relatively compact, subordinated with order $p<1$, locally subordinated. For different kinds of perturbations we prove theorems on preservation properties of the completeness, basisness or unconditional basisness of the root vectors, as well as comparision theorems for the eigenvalue distribution of the operators $A$ and $T$.