  We analyze the perturbations $T+B$ of a selfadjoint operator $T$ in a Hilbert
space $H$ with discrete spectrum $\{t_k \}$, $T \phi_k = t_k \phi_k$, as an
extension of our constructions in arXiv: 0912.2722 where $T$ was a harmonic
oscillator operator. In particular, if $t_{k+1}-t_k \geq c k^{\alpha - 1},
\quad \alpha > 1/2$ 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 $H$.
