Topics in multivariable operator theory

本课题关注多变量算子理论的某些方面的进一步发展，特别是它与几个复杂变量的函数理论的联系。该理论的基本观点是，人们可以通过考虑解析函数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，这一观点在单变量的背景下得到了广泛的发展，在复函数理论、算子理论和泛函分析之间建立了深刻的联系；冯·诺伊曼不等式和Sz。-纳吉膨胀定理是这一领域的基本成果。本项目将研究的具体问题包括冯·诺伊曼不等式在若干变量中的失效，解析函数的泛函理论方面，实现为由算子元组生成的传递函数，以及用“非交换谱测度”（即算子系统上的正泛函）来分析正实部函数。一个统一的主题将是几个变量中的复合算子理论，特别是紧性问题。这个项目属于被称为算子理论的数学分支。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。它在工程上有许多应用，例如在飞机和航天器的鲁棒控制系统的设计中；声散射数据分析；最近在线性矩阵不等式的研究中，它隐藏在许多优化问题背后。相反，工程和数学物理中的问题对算子理论提出了新的问题；与该项目相关的是对复合材料静电特性的描述。

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.