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A note on the boundary of the joint numerical range. (English) Zbl 1429.15019
Let $$A_{1},\dots,A_{n}$$ be self-adjoint operators on a complex Hilbert space $$H$$. The joint numerical range of the $$n$$-tuple $$A=(A_{1},\dots,A_{n})$$ is the set $$W(A)=\{(\left\langle A_{1}x,x\right\rangle\dots,\left\langle A_{n}x,x\right\rangle ):x\in H,\left\Vert x\right\Vert =1\}$$.
The author gives results about the boundary of $$W(A)$$ and its closure. It is shown that $$W(A)$$ is closed if and only if $$W_{e}(A)\subseteq W(A)$$ and every point in $$W_{e}(A)$$ is a star centre of $$W(A)$$, where $$W_{e}(A)$$ is the joint essential numerical range of $$A$$.

##### MSC:
 15A60 Norms of matrices, numerical range, applications of functional analysis to matrix theory 47A12 Numerical range, numerical radius
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