Abstract
For a bounded linear operator T in a Hilbert space denote by ɛ(T) the set of the extreme points of its numerical range W(T), and by K(T) the subset of the points of ɛ(T) which lie on the boundary of a disc being a spectral set for T in the sense of J. von Neumann. Each corner of W(T) belongs to K(T). This implies: K(T) is dense in ɛ(T) if T is compact and the convex hull of its spectrum contains W(T). If Re T≥0 and o ε ɛ(T) then a necessary and sufficient condition is given that O is a eigenvalue of T. This is applied to show that W(T) ∩ ɛ(T) is a subset of the point spectrum of T for a class of operators containing the hyponormal operators.
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Literatur
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Förster, KH. Über Extremalpunkte des numerischen Wertebereichs eines linearen Operators. Manuscripta Math 1, 1–7 (1969). https://doi.org/10.1007/BF01171130
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DOI: https://doi.org/10.1007/BF01171130