Protecting points from operator pencils

TL;DR

This paper classifies the spectral sets formed by the union of spectra of self-adjoint operator pencils, showing they are complements of at most countable discrete subsets of the real line.

Contribution

It provides a complete classification of spectral sets generated by operator pencils with bounded, non-negative, non-zero operators, identifying them as complements of discrete subsets of the real line.

Findings

Spectral sets are exactly the complements of discrete subsets of .

These sets contain no accumulation points.

The classification applies to all such 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.