Topics in multivariable operator theory

这个项目是关于多变量算子理论某些方面的进一步发展，特别是它与多复变函数论的联系。该理论的基本观点是，通过考虑解析函数对矩阵或运算符输入的作用，而不仅仅是标量输入，可以有益地研究解析函数。在过去的半个世纪里，这一观点在单变量背景下得到了广泛的发展，在复函数论、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-Nagy膨胀定理是这一领域的基本结果。这个项目中将研究的特殊问题包括：冯·诺伊曼在多变量中的不成立，解析函数的函数论方面，实现为由算子元组生成的传递函数的方面，以及关于“非对易谱度量”的正实部函数的分析，也就是算子系统上的正泛函。一个统一的主题将是几个变量的复合算符理论，特别是紧致性问题。该项目属于数学的一个分支，称为算符理论。它最初是作为量子力学的数学语言发展起来的，在过去的一个世纪里，它已经扩展到影响到科学和数学的许多其他领域。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；在声散射数据的分析中；以及最近在许多优化问题背后的线性矩阵不等式的研究中。相反，工程和数学物理中的问题在算符理论中提出了新的问题；与该项目相关的一个问题是对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.