Journal of Operator Theory

Volume 85, Issue 2, Spring 2021 pp. 383-389.

Protecting points from operator pencils

Authors: Albrecht Seelmann (1), Matthias Taufer (2), Kresimir Veseli (3)
Author institution: (1) Fakultaet fuer Mathematik, Technische Universitaet Dortmund, Dortmund, D-44221, Germany
(2) School of Mathematical Sciences, Queen Mary University of London, London, E1 4NS, U.K.
(3) Fakultaet fuer Mathematik und Informatik, Fernuniversitaet Hagen, Hagen, D-58084, Germany

Summary: We classify all sets of the form $\bigcup\limits_{t\in\mathbb{R}}\mathrm{spec}(A+tB)$ 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.

DOI: http://dx.doi.org/10.7900/jot.2019aug23.2248
Keywords: union of spectra, operator pencil, homogeneous operator

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