On the basis property of root vectors of a~perturbed self-adjoint operator

A. A. Shkalikov

Trudy Matematicheskogo Instituta imeni V.A. Steklova, Function theory and differential equations, Tome 269 (2010), pp. 290-303

Résumé

We study perturbations of a self-adjoint operator $T$ with discrete spectrum such that the number of its points on any unit-length interval of the real axis is uniformly bounded. We prove that if $\|B\varphi_n\|\le\mathrm{const}$, where $\varphi_n$ is an orthonormal system of eigenvectors of the operator $T$, then the system of root vectors of the perturbed operator $T+B$ forms a basis with parentheses. We also prove that the eigenvalue-counting functions of $T$ and $T+B$ satisfy the relation $|n(r,T)-n(r,T+B)|\le\mathrm{const}$.