Opuscula Math. 35, no. 3 (2015), 279-285
http://dx.doi.org/10.7494/OpMath.2015.35.3.279

Opuscula Mathematica

# Hildebrandt's theorem for the essential spectrum

Janko Bračič
Cristina Diogo

Abstract. We prove a variant of Hildebrandt's theorem which asserts that the convex hull of the essential spectrum of an operator $$A$$ on a complex Hilbert space is equal to the intersection of the essential numerical ranges of operators which are similar to $$A$$. As a consequence, it is given a necessary and sufficient condition for zero not being in the convex hull of the essential spectrum of $$A$$.

Keywords: essential spectrum, essential numerical range, Hildebrandt's theorem.

Mathematics Subject Classification: 47A10, 47A12.

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• Janko Bračič
• University of Ljubljana, IMFM, Jadranska ul. 19, SI-1000 Ljubljana, Slovenia
• Cristina Diogo
• Instituto Universitário de Lisboa, Departamento de Matemática, Av. das Forças Armadas, 1649-026 Lisboa, Portugal
• Center for Mathematical Analysis, Geometry, and Dynamical Systems, Mathematics Department, Instituto Superior Técnico, Universidade de Lisboa, Av. Rovisco Pais, 1049-001 Lisboa, Portugal