Summary
The classical proof that a separated Gershgorin disc contains an eigenvalue uses a continuity argument which is not applicable to Bauers generalized Gershgorin discs. In this paper the concept of positivity, generalized to complex vector spaces, is used to establish Gershgorins result and to improve the corresponding result for Bauers generalization.
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Bauer, F.L.: On the field of values subordinate to a norm. Numer. Math.4, 103–113 (1962/63)
Bauer, F.L.: Fields of values and Gershgorin discs. Numer. Math.12, 91–95 (1968)
Deutsch, E., Zenger, C.: On Bauer's generalized Gershgorin discs. Numer. Math.24, 63–70 (1975)
Gershgorin, S.: Über die Abgrenzung der Eigenwerte einer Matrix. Izv. Akad. Nauk Armjan. SSSR Ser. Mat.7, 749–754 (1931)
Krein, M.G., Rutman, M.A.: Linear operators leaving invariant a cone in a Banach space. Uspehi Mat. Nank (N.S.)3, no. 1 (23), 3–95 (1948) (Russian)
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Dedicated to Professor F.L. Bauer on the occasion of his 60th birthday
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Zenger, C. Positivity in complex spaces and its application to Gershgorin discs. Numer. Math. 44, 67–73 (1984). https://doi.org/10.1007/BF01389756
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DOI: https://doi.org/10.1007/BF01389756