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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)

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