Topics in multivariable operator theory

这个项目关注多变量算子理论某些方面的进一步发展，特别是它与多复变量函数理论的联系。该理论的基本观点是，人们可以通过考虑它们对矩阵或算子输入的作用而不仅仅是标量输入来有益地研究解析函数。在过去的半个世纪里，这种观点在单变量环境中得到了广泛的发展，建立了复变函数理论，算子理论和泛函分析之间的深刻联系;冯诺依曼不等式和Sz。Nagy伸缩定理是这方面的基本结果。 特别的问题，将在这个项目中进行调查，包括失败的冯诺依曼不等式在几个变量，功能理论方面的解析函数实现的传递函数所产生的运营商元组，并分析功能的积极真实的部分的“非交换谱措施”，即积极的泛函对运营商系统。一个统一的主题将是理论的复合算子在几个变量，特别是紧凑性问题。该项目属于分支的数学称为算子理论。最初是作为量子力学的数学语言发展起来的，在上个世纪，它已经扩展到影响许多其他科学和数学领域。它在工程中有许多应用，例如在飞机和航天器的鲁棒控制系统的设计中;声散射数据的分析;以及最近在线性矩阵不等式的研究中，这是许多优化问题的背后。相反，工程和数学物理的问题在算子理论中提出了新的问题;与该项目相关的一个问题是复合材料的静电特性的描述。

This project is concerned with the further development of certain aspects of multivariable operator theory, especially its connection with function theory in several complex variables. The basic point of view of the theory is that one can profitably study analytic functions by considering their action on matrix or operator inputs, rather than just scalar inputs. Over the last half-century this viewpoint has been extensively developed in the one-variable setting, establishing deep connections between complex function theory, operator theory, and functional analysis; von Neumann's inequality and the Sz.-Nagy Dilation theorem are fundamental results in this area. Particular problems that will be investigated in this project include the failure of von Neumann's inequality in several variables, function-theoretic aspects of analytic functions realized as transfer functions generated by operator tuples, and the analysis of functions of positive real part in terms of "noncommutative spectral measures," that is, positive functionals on operator systems. A unifying theme will be the theory of composition operators in several variables, especially compactness questions.The project belongs to the branch of mathematics known as Operator Theory. Originally developed as the mathematical language of quantum mechanics, over the last century it has expanded to influence many other areas of science and mathematics. It has found many applications in engineering, for example in the design of robust control systems for aircraft and spacecraft; the analysis of acoustical scattering data; and more recently in the study of linear matrix inequalities, which lie behind many optimization problems. Conversely, problems in engineering and mathematical physics have raised new questions in operator theory; one related to the project is the description of the electrostatic properties of composite materials.