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

We analyze the perturbations + of a selfadjoint operator in a Hilbert space with discrete spectrum . In particular, if (+1) - a parts per thousand yen (- 1), > 1/2 and then the system of root vectors of + , eventually eigenvectors of geometric multiplicity 1, is an unconditional basis in (Theorem 6). Under the assumptions (with and fixed) and a Riesz system { } of projections on invariant subspaces of + , Rank a parts per thousand currency sign , is constructed (Theorem 3).