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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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