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
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
- Communicated by Aurelian Gheondea.
- Received: 2014-08-01.
- Accepted: 2014-10-01.
- Published online: 2014-12-15.
