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Convexity of the joint numerical range. (English) Zbl 0945.15019
Let \(A=(A_1,\ldots,A_m)\) be an \(m\)-tuple of \(n\times n\) Hermitian matrices. For \(1\leq k\leq n,\) the \(k\)th joint numerical range of \(A\) is defined as \[ W_k(A)=\{(\text{tr}(X^\star A_1X),\dots,\text{tr}(X^\star A_mX))\mid X\in{\mathbb{C}}^{n\times k},X^\star X=I_k\}. \]
The authors pose a number of problems, e.g. characterize maximal linearly independent convex families for the \(k\)th numerical range. They give some \(A\) for which \(W_k(A)\) is not convex and they show that there are independent families \(A\) with \(m=2k(n-k)+1\) such that \(W_k(A)\) is convex. The key idea here is to view \(W_k(A)\) as the image of all rank \(k\) projections under some linear map.
The paper also deals with a number of related problems.
Reviewer: E.Ellers (Toronto)

15A60 Norms of matrices, numerical range, applications of functional analysis to matrix theory
15B57 Hermitian, skew-Hermitian, and related matrices
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