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On the generalized numerical range. (English) Zbl 0648.15017
Let \(A_ k\), \(k=1,...,m\), be \(n\times n\) Hermitian matrices. Let \(f: {\mathbb{C}}\) \(n\to {\mathbb{R}}^ m \)have components \(f\quad k(x)=x\quad HA_ kx,\) \(k=1,...,m\), \(W(A_ 1,...,A_ k)=\{f(x): \| x\| =1\}.\) It is known that W is convex when \(n\geq 3\) and \(m=3\), and W is not convex in general when \(m>3\). The authors give geometric proofs of these results and study the geometry of W.
Reviewer: N.M.Zobin

MSC:
15A60 Norms of matrices, numerical range, applications of functional analysis to matrix theory
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