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 $\{t_k \}$, $T ϕ_k = t_k ϕ_k$, as an extension of our constructions in arXiv: 0912.2722 where $T$ was a harmonic oscillator operator. In particular, if $t_{k+1}-t_k \geq c k^{α- 1}, \quad α> 1/2$ and $\| B ϕ_k \| = o(k^{α- 1})$ then the system of root vectors of $T+B$, eventually eigenvectors of geometric multiplicity 1, is an unconditional basis in $H$.

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