Localization of the spectrum of certain non-self-adjoint operators

M. M. Gekhtman

Matematičeskie zametki, Tome 3 (1968) no. 4, pp. 415-419.

Résumé

Let the self-adjoint operator $A$ and the bounded operator $B$ be specified in Hilbert space $\mathscr H$. We let denote the spectral family of the operator $A$. If $\|(E-E_N)B\|^2+E_{-N}B\|^2\to 0$, then in the complex plane $z=\sigma+\tau$ there will exist the curve $|\tau|=f(\sigma)$, $\lim f(\sigma)=0$ for $\sigma\to\pm\infty$ such that the entire spectrum of the operator $A+B$ lies within the region $|\tau|\le f(\sigma)$. In particular, the condition of the theorem will be satisfied when $B$ is a completely continuous operator.