Linearity and weak convergence on the boundary of numerical range
Linearity and weak convergence on the boundary of numerical range
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DOI：
10.1017/s1446788700025714
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发表时间：
1983-10
期刊：
影响因子：
--
通讯作者：
K. C. Das;B. Craven
中科院分区：
文献类型：
--
作者：
K. C. Das;B. Craven
摘要 Stampfli 和 Embry 证明，当且仅当对应于算子数值范围的一个点是线性的时，该点才是极值点。此处对此进行概括，以表明当且仅当相应的一组序列形成线性空间时，数值范围的闭合点才是极端的。对Das和Garske关于在数值范围闭包的未达到的极值点处弱收敛到零的定理给出了更具几何意义的替代证明。结果表明，对于位于其边界上的线段上的数值范围的孤立极值点也成立。进一步地，在与数值范围边界上的线段上的点对应的弱收敛序列的弱极限的范数上获得界限。
Abstract Stampfli and Embry have shown that a point of the numerical range of an operator is extreme if and only if a set of vectors corresponding to it is linear. This is generalized here to show that a point of the closure of the numerical range is extreme if and only if a corresponding set of sequences forms a linear space. A more geometric alternative proof is given for a theorem of Das and Garske concerning weak convergence to zero at the unattained extreme points of the closure of the numerical range.The result is shown to hold also for lone extreme points of the numerical range which lie on line segments on its boundary. Further, a bound is obtained on the norm of the weak limit of the weakly convergent sequences corresponding to points on a line segment on the boundary of numerical range.