论文标题
轨道封闭的$ c $ numerical范围和多数化的凸面
Convexity of the orbit-closed $C$-numerical range and majorization
论文作者
论文摘要
我们介绍并研究了轨道封闭的$ c $ numerical范围，这是Dirr和Vom Ende的$ c $ trace-class介绍的$ c $ numerical范围的自然修改。我们锁定轨道$ c $ numerical的范围是对其轨道的保守修改，因为这两组具有相同的关闭，甚至在$ c $是有限的排名时，甚至重合。由于Dirr和Vom Ende关于$ C $数字范围的结果仅取决于其关闭，因此我们的轨道封闭的$ C $数字范围继承了这些属性，但我们也建立了更多。 对于$ c $ selfadexhind，dirr和vom ende只能证明其$ c $ numerical范围的关闭是凸的，并询问它是否是凸的而无需关闭。我们建立了轨道封闭的$ c $ numerical范围的selfAdexhixhexhiendexical $ c $范围，而无需通过在大分化方面提供特征，从而释放了多个结果的大门，这些结果概括了$ c $ numererical-numererical范围的属性，以有限的尺寸或$ c $ $ c $具有有限的等级。在操作员的相当特殊的假设下，我们还显示了$ c $ numerical范围是凸的，从而为Dirr和Vom Ende提出的问题提供了部分答案。
We introduce and investigate the orbit-closed $C$-numerical range, a natural modification of the $C$-numerical range of an operator introduced for $C$ trace-class by Dirr and vom Ende. Our orbit-closed $C$-numerical range is a conservative modification of theirs because these two sets have the same closure and even coincide when $C$ is finite rank. Since Dirr and vom Ende's results concerning the $C$-numerical range depend only on its closure, our orbit-closed $C$-numerical range inherits these properties, but we also establish more. For $C$ selfadjoint, Dirr and vom Ende were only able to prove that the closure of their $C$-numerical range is convex, and asked whether it is convex without taking the closure. We establish the convexity of the orbit-closed $C$-numerical range for selfadjoint $C$ without taking the closure by providing a characterization in terms of majorization, unlocking the door to a plethora of results which generalize properties of the $C$-numerical range known in finite dimensions or when $C$ has finite rank. Under rather special hypotheses on the operators, we also show the $C$-numerical range is convex, thereby providing a partial answer to the question posed by Dirr and vom Ende.
