Convexity of the Joint Numerical Range: Topological and Differential Geometric Viewpoints

Eugene Gutkin, Edmond Jonckheere, and Michael Karow

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Submission date: 02. Jan. 2003 (revised version: June 2004)
Pages: 37
published in: Linear algebra and its applications, 376 (2004), p. 143-171 
DOI number (of the published article): 10.1016/j.laa.2003.06.011
Bibtex
Keywords and phrases: joint numerical range, support function, highest eigenvalue
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Abstract:
We show that the outer boundary of the joint numerical range of any number, m, of hermitian matrices is convex if the multiplicity of the largest eigenvalue of the associated hermitian matrix is constant. Thus for m>3 the problem of convexity of the joint numerical range is essentially topological.

Furthermore, our sufficient condition for convexity of the joint numerical range is also necessary for its stable convexity.