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Updated: Jan 9, 2026

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Generation and Coherent Control of Pulsed Quantum Frequency Combs
Published on: June 8, 2018
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托普利茨-合约运营商组的矩阵凸度和单元功率扩展
1Department of Mathematics and Statistics, University of Regina, Regina, Saskatchewan S4S 0A2 Canada.
概括
研究人员为收缩线性运算符重新构建了单元扩展定理. 这导致了对"Toeplitz-contractive"运算符d-tuples的新定义,在运算符理论中实现了新的距离测量.
科学领域:
- 运算子理论 运算子理论
- 功能分析是一种功能分析.
- 多变量运算子理论
背景情况:
- 这项研究建立在已确定的PR定理之上. 哈尔莫斯关于希尔伯特空间中的合约线性演算子的单元扩展.
- 利用T. Ando和L. Gurvits的基础工作来扩展现有概念.
研究的目的:
- 为收缩的希尔伯特空间运算符的d-tuples重新制定哈尔莫斯定理.
- 引入和描述一类新的经营者,称为"Toeplitz合同"经营者.
- 在多变量运算符理论中定义新的距离尺度.
主要方法:
- 使用对运算符d-tuples的矩阵正值条件,重新构造单元扩展定理.
- 为新定义的Toeplitz-contractivity条件开发了一种表征.
- 将规范,数值半径和光谱半径的概念概括为运算符的d-tuples.
主要成果:
- 引入了满足矩阵正性条件的运算符的"Toeplitz合约"d-tuples.
- 定义了d-tuples的新非对称距离尺度,将经典运算符规范概括.
- 确定Toeplitz合同运营商构成一个非交换的凸集.
- 确定矩阵凸集合的缩放常数,与d复杂维度的伸缩单位圆有关.
结论:
- 该研究成功地将哈尔莫斯的单元扩展定理扩展到一个d变量设置.
- 引入了新的操作员类和距离测量,为多变量操作员理论提供了新的工具.
- 这些发现提供了对托普利茨合约运算子和相关凸集的几何性质的洞察.
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