Perturbations of self-adjoint and normal operators with discrete spectrum

A. A. Shkalikov

Trudy Matematicheskogo Instituta imeni V.A. Steklova, Tome 71 (2016) no. 5, pp. 907-964

Résumé

The spectral properties of operators of the form $A=T+B$ are analyzed, where $B$ is a non-symmetric operator subordinate to a self-adjoint or normal operator $T$. The different definitions of perturbations with respect to $T$ are considered: completely subordinated, subordinate with order $p1$, locally subordinate. Analogues of these types of perturbations are considered also for operators defined in terms of quadratic forms. For perturbations of different types, series of statements on the completeness property of the root vectors of the operator and on the basis or unconditional basis property are proved. The spectra of the operators $T$ and $T+B$ are compared as well. A survey of research in this area is presented. Bibliography: 89 titles.

Détail

Keywords: perturbations of linear operators, resolvent estimates, conditions for local subordination, sums of the quadratic forms of operators, unconditional bases, Riesz bases, Abel–Lidskii summability method.
Mots-clés : conditions for $p$-subordination