Topics in Dilation Theory

膨胀理论专题

• 批准号: 0457504
• 负责人: Scott McCullough
• 金额: $ 5.16万
• 依托单位: University of Florida
• 依托单位国家: 美国
• 项目类别: Standard Grant
• 财政年份: 2005
• 资助国家: 美国
• 起止时间: 2005-05-15 至 2008-04-30
• 项目状态: 已结题
• 来源: https://www.nsf.gov/awardsearch/showAward?AWD_ID=0457504&HistoricalAwards=false
• 关键词: Topics; Dilation; Theory

The Sz.-Nagy Dilation Theorem, which models a contraction operator on Hilbert space as the adjoint of an isometry restricted to an invariant subspace has had a profound influence on operator theory and its applications. Indeed, dilation theorems, which represents an operator (or algebra of operators) on Hilbert space from a specialized class as a coherent part of a nice operator (or algebra from the class are an integral part of operator theory and connect to a broad range of mathematics and have significant applications in physics, systems theory, and engineering. This project pursues three related lines of investigation. Namely, dilation theoretic aspects of multi-variable operator theory in both the commutative and non-commutative setting and single-variable operator theory on multiply connected domains. There are two objectives in the non-commutative case. First to establish that convexity in this setting must take a very simple form; and second to begin the concomitant development of a theory of non-commutative real algebraic geometry. The study of the dilation theory and completely positive maps associated to multiply connected domains has a long tradition in operator theory and an aim of this project is study a problem known as rational dilations for domains of genus three or larger.(A disc with g discs removed is a domain of genus g.) This work is expected to make connections with the theory of theta functions and Riemann surfaces. successfully used in the genus two case include the theory of theta functions, Riemann surfaces. The work on commutative multi-variable operator theory, more accurately described as the study of non-self-adjoint operator algebras (with some added structure), has as an objective the extension of familiar operator theoretic results for $H^\infty$ of the unit disc. Operator theory has a history of rich interplay with engineering and physics as well as other vital areas of mathematics including complex function theory and algebraic geometry. Originally developed as a tool to study integral and differential equations arising in physics, operator theory, operator algebras, and operator systems play an important role in modern quantum physics and fundamentals of operator theory are now basic technology in systems theory which in turn has important applications in image processing and control theory - the mathematics behind automatic controllers such as autopilots. The proposed work builds on operatortheoretic ideas with a view toward applications to systems theory, control theory, operator algebras, and function theory. The investigation also connects with non-commutative algebraic geometry and Linear Matrix Inequalities or LMIs. Many engineering problems can be formulated as matrix inequalities in which the variables are noncommutative (xy is not the same as yx). Moreover, often convexity is either present, as is the case with LMIs, or desirable. This project will pursue the theme that convexMIs can be converted to LMIs.

Sz.- Nagy伸缩定理将Hilbert空间上的压缩算子模拟为约束于不变子空间的等距算子的伴随，对算子理论及其应用产生了深远的影响。事实上，扩张定理，它表示希尔伯特空间上的一个算子（或算子代数）从一个专门的类作为一个好的算子（或代数从类的相干部分是算子理论的一个组成部分，并连接到广泛的数学，并在物理学，系统理论和工程有重要的应用。本项目遵循三条相关的调查路线。即，膨胀理论方面的多变量算子理论在交换和非交换设置和单变量算子理论在多连通域。在非交换的情况下有两个目标。第一，在这种情况下，凸性必须采取一种非常简单的形式;第二，开始伴随着非交换真实的代数几何理论的发展。与多连通域相关的扩张理论和完全正映射的研究在算子理论中有着悠久的传统，本项目的目的是研究亏格为3或更大的域的有理扩张问题。（去掉g个圆盘的圆盘是g属的域。这项工作预计将与理论的theta函数和黎曼曲面。在亏格中成功应用的两种情况包括theta函数理论、Riemann曲面。工作交换多变量算子理论，更准确地描述为研究非自伴算子代数（有一些增加的结构），有作为一个目标的扩展熟悉的算子理论结果为$H^\infty$的单位盘。算子理论与工程和物理以及其他重要的数学领域（包括复变函数论和代数几何）有着丰富的相互作用的历史。最初是作为研究物理学中的积分和微分方程的工具而开发的，算子理论，算子代数和算子系统在现代量子物理学中发挥着重要作用，算子理论的基础现在是系统理论中的基础技术，反过来又在图像处理和控制理论中有重要的应用-自动控制器背后的数学，如自动驾驶仪。拟议的工作建立在Operatortheorical的思想，以期对系统理论，控制理论，算子代数和函数理论的应用。该研究还与非交换代数几何和线性矩阵不等式或线性矩阵不等式有关。许多工程问题可以用矩阵不等式表示，其中变量是非交换的（xy与yx不同）。 此外，通常凸性要么是存在的，如LMI的情况，要么是期望的。这个项目将追求的主题，凸MI可以转换为LMI。