Abstract
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 α - 1, α > 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).
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Adduci, J., Mityagin, B. Root System of a Perturbation of a Selfadjoint Operator with Discrete Spectrum. Integr. Equ. Oper. Theory 73, 153–175 (2012). https://doi.org/10.1007/s00020-012-1967-7
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DOI: https://doi.org/10.1007/s00020-012-1967-7