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Convexity of the orbit-closed C-numerical range and majorization
Linear and Multilinear Algebra ( IF 1.1 ) Pub Date : 2021-04-27 , DOI: 10.1080/03081087.2021.1895702
Jireh Loreaux 1 , Sasmita Patnaik 2
Affiliation  

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 self-adjoint, 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 self-adjoint 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.



中文翻译:

闭轨道C数值范围的凸性和主化

我们介绍并研究了轨道闭合的C数值范围,这是Dirr和vom Ende为C跟踪类引入的算子的C数值范围的自然修改。我们的闭轨C数值范围是对其的保守修改,因为这两个集合具有相同的闭角,并且当C为有限秩时甚至重合。由于Dirr和vom Ende关于C数值范围的结果仅取决于其闭合,因此我们的轨道闭合C数值范围继承了这些属性,但我们还要建立更多的性质。对于C自伴而言,Dirr和vom Ende仅能证明其C的关闭-数值范围是凸的,并询问它是否是凸的,而无需进行闭合。我们通过提供主要化方面的特征来建立自伴C的轨道闭合C数值范围的凸度,而无需采取闭合,通过为大量结果开辟大门,这些结果概括了已知的C数值范围的性质有限尺寸或当C具有有限等级时。在关于算子的相当特殊的假设下,我们还证明了C数值范围是凸的,从而为Dirr和vom Ende提出的问题提供了部分答案。

更新日期:2021-04-27
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