Root System of a Perturbation of a Selfadjoint Operator with Discrete Spectrum

We analyze the perturbations T?+?B of a selfadjoint operator T in a Hilbert space H with discrete spectrum ${{ t_k}, T phi_k = t_k phi_k}$ . In particular, if t _k+1 ? t _k ?? ck ^{?? - 1}, ?? > 1/2 and ${| B phi_k | = o(k^{alpha - 1})}$ then the system of root vectors of T?+?B, eventually eigenvectors of geometric multiplicity 1, is an unconditional basis in H (Theorem 6). Under the assumptions ${t_{k+p} - t_k geq d > 0, forall k}$ (with d and p fixed) and ${| B phi_k | rightarrow 0}$ a Riesz system {P _k } of projections on invariant subspaces of T?+?B, Rank P _k ?? p, is constructed (Theorem 3).