Protecting points from operator pencils

Abstract

We classify all sets of the form ${\bigcup }_{t\in \mathbb{R}}\mathrm{spec}(A+tB)$ where $A$ and $B$ are self-adjoint operators and $B$ is bounded, non-negative, and non-zero. We show that these sets are exactly the complements of discrete subsets of $\mathbb{R}$, that is, of at most countable subsets of $\mathbb{R}$ that contain none of their accumulation points.

6 pages; a counterexample for sign-indefinite perturbations and some references have been added, some editorial changes

References in corpus (1)

• Simplicity of eigenvalues in Anderson-type models