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

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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References ▼
  1. [1] S.K. Berberian, The Weyl spectrum of an operator, Indiana Univ. Math. J. 20 (1970/1971), 529-544.
  2. [2] F.F. Bonsall, J. Duncan, Numerical ranges of operators on Normed spaces and of elements of normed algebras, London Mathematical Society Lecture Note Series, No. 2, Cambridge University Press, London-New York, 1971.
  3. [3] F.F. Bonsall, J. Duncan, Numerical ranges II, London Mathematical Society Lecture Notes Series, No. 10, Cambridge University Press, New York-London, 1973.
  4. [4] R.G. Douglas, Banach algebra techniques in operator theory, Academic Press, 1972.
  5. [5] K.E. Gustafson, D.K.M. Rao, Numerical Range, Springer-Verlag, New York, 1997.
  6. [6] P.R. Halmos, A Hilbert space problem book, Springer-Verlag, New-York, 1982.
  7. [7] S. Hildebrandt, Über den Numerischen Wertebereich eines Operators, Math. Ann. 163 (1966), 230-247.
  8. [8] G.J. Murphy, T.T. West, Spectral radius formulae, Proc. Edinburgh Math. Soc. 22 (1979), 271-275.
  9. [9] J.G. Stampfli, J.P. Williams, Growth conditions and the numerical range in a Banach algebra, Tohoku Math. Journ. 20 (1968), 417-424.
About the authors ▼
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
About the article ▼
Communicated by Aurelian Gheondea.
Received: 2014-08-01.
Accepted: 2014-10-01.
Published online: 2014-12-15.
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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