Opuscula Math. 35, no. 3 (2015), 279-285

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
  • Received: 2014-08-01.
  • Accepted: 2014-10-01.
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Cite this article as:
Janko Bračič, Cristina Diogo, Hildebrandt's theorem for the essential spectrum, Opuscula Math. 35, no. 3 (2015), 279-285, http://dx.doi.org/10.7494/OpMath.2015.35.3.279

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