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

Abstract

We analyze the perturbations $T+B$ of a selfadjoint operator $T$ in a Hilbert space $H$ with discrete spectrum $\{t_k\}$, $T{\varphi }_k=t_k{\varphi }_k$, as an extension of our constructions in arXiv: 0912.2722 where $T$ was a harmonic oscillator operator. In particular, if $t_{k+1}-t_k\ge c{k}^{\alpha -1},\alpha >1/2$ and $\Vert B{\varphi }_k\Vert =o(k^{\alpha -1})$ then the system of root vectors of $T+B$, eventually eigenvectors of geometric multiplicity 1, is an unconditional basis in $H$.