Operator inequalities, reproducing kernels, and invariant subspaces

算子不等式、再现核和不变子空间

• 批准号: 0070451
• 负责人: Stefan Richter
• 金额: $ 18.6万
• 依托单位: University of Tennessee Knoxville
• 依托单位国家: 美国
• 项目类别: Continuing grant
• 财政年份: 2000
• 资助国家: 美国
• 起止时间: 2000-06-15 至 2004-05-31
• 项目状态: 已结题
• 来源: https://www.nsf.gov/awardsearch/showAward?AWD_ID=0070451&HistoricalAwards=false
• 关键词: Operator; inequalities; reproducing; kernels; invariant

AbstractRichter/SundbergRichter and Sundberg will continue their study of linear operators on Hilbert spaces modelled by multiplication operators on spaces of analytic functions, in both one and several variables. The study of such operators has a long history and has led to much progress in operator theory and complex analysis. For instance the unilateral shift, which is modelled by multiplication by z on the Hardy space of the unit disc, has been used to enhance our knowledge of contraction operators. Other classes of operators characterized by operator inequalities may be studied using operators modelled by multiplication by z on other Hilbert spaces of analytic functions. Such spaces have associated with them evaluation functionals called reproducing kernels, and there is an intimate connection between properties of such a kernel and properties of the multiplication operators on the corresponding space. An important classical example where this correspondence has been exploited is Nevanlinna-Pick interpolation. Recent work of several researchers has extended the applicability of the ideas behind Nevanlinna-Pick interpolation and introduced other types of inequalities on operators and reproducing kernels that seem to be very fruitful. Richter and Sundberg will investigate the connection between operator inequalities, reproducing kernels, and the structure of the lattice of invariant subspaces of multiplication operators. The proposed work involves ideas and problems from several areas of pure and applied mathematics. Operator Theory, which may be thought of as an infinite dimensional version of linear algebra, grew out of ideas used to study certain partial differential equations arising in physics in the 1800's, and became increasingly important with the advent of Quantum Mechanics in the twentieth century. Complex Analysis is a subject with a long and distinguished history, and remains a very active and broad area of research. These two areas have had a very fruitful interaction throughout this century, owing to the fact that some interesting and useful operators can be modelled by natural operations on spaces of analytic functions. At least since the 1960's it has been realized that a series of related results concerning certain of these operators are of importance in the study of Control Theory, an area of importance in electrical engineering and other practical applications. Among these results are the Beurling-Lax Theorem on invariant subspaces, the Nevanlinna-Pick Interpolation Theorem, and its close relative the Commutant Lifting Theorem. Work by a number of researchers since the 1980's has shown that the circle of ideas concerned with these results are applicable to a much wider class of objects than had previously been realized. This has resulted in a much improved understanding of the underlying mathematical systems

Richter和Sundberg将继续他们的研究线性算子的希尔伯特空间模拟乘法算子空间的解析函数，在一个和几个变量。对这类算子的研究有着悠久的历史，并在算子理论和复分析方面取得了很大的进展。例如，单边移位，这是仿照乘以z上的哈代空间的单位盘，已被用来提高我们的知识的收缩运营商。其他类型的算子的特点是算子不等式可以研究使用的运营商的乘法由z建模的其他希尔伯特空间的解析函数。这样的空间与它们相关联的评价泛函称为再生核，并且这样的核的性质与相应空间上的乘法算子的性质之间存在密切联系。一个重要的经典例子，这种对应关系已被利用是Nevanlinna-Pick插值。最近几个研究人员的工作扩展了Nevanlinna-Pick插值背后的思想的适用性，并引入了其他类型的算子和再生核的不等式，这些不等式似乎非常富有成效。Richter和Sundberg将研究算子不等式、再生核和乘法算子的不变子空间格的结构之间的联系。 拟议的工作涉及的想法和问题，从几个领域的纯数学和应用数学。算子理论，这可能被认为是一个无限维版本的线性代数，成长的想法用来研究某些偏微分方程在物理学中出现在1800年，并成为日益重要的量子力学的出现在二十世纪。 复变函数分析是一个有着悠久而杰出历史的学科，并且仍然是一个非常活跃和广泛的研究领域。 这两个领域有一个非常富有成效的互动在整个世纪，由于事实上，一些有趣的和有用的运营商可以模拟自然操作空间的解析函数。至少从20世纪60年代以来，人们已经认识到，关于某些算子的一系列相关结果在控制理论的研究中是重要的，控制理论是电气工程和其他实际应用中的一个重要领域。这些结果中有Beurling-Lax定理 在不变子空间上，Nevanlinna-Pick插值定理，及其近亲交换提升定理。自20世纪80年代以来，许多研究人员的工作表明，与这些结果有关的思想圈适用于比以前认识到的更广泛的对象。这导致了对基本数学系统的理解大大提高