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).