Mathematical Sciences: Subnormal Operators and Related Topics

算子论是现代分析的中心学科。它起源于二十世纪初对数学物理和偏微分方程式的研究。因此，可以看到，平衡论、振动、量子理论等中的许多物理问题都可以通过对现象进行建模的积分方程式来有效地研究。因此，从希尔伯特、冯·诺伊曼和其他巨人的丰富头脑中，算子理论的主题已经成长为此类研究的中心位置，在核心数学中也是如此。这种方法的核心是深入研究算子的谱及其不变子空间。对于所谓的自伴算子，这一理论现在是整个分析的标准技术，而谱定理为所有这类算子提供了必要的基础。虽然不是很完整，但谱理论被广泛地理解为自伴情况的广泛推广；即，“正常”算符。因此，目前对这一结构理论的研究前沿在于非正态理论。Conway教授长期以来一直是研究次正规算子的领导者，次正规算子是应用中出现的一类广泛的算子，对于这些算子，有足够的剩余正规信息来尝试更深入地了解它们的内在结构。最近的结果，如次正规算子是自反的和有不变子空间的事实，增加了对它们的谱性质的研究的动力。由于在复变函数论、微分几何和逼近理论中出现的许多显式算子都是不正常的，因此存在着广泛的发展中的相互作用。Conway教授将研究Hardy空间上的作用中自然出现的次正规算子，从而探索与函数论的相互作用。他还将研究次正规算子的内部结构理论，特别考虑酉等价性、丛移以及与一般算子理论的关系。

Operator theory is a central discipline in Modern Analysis. Its origins lie in the study of mathematical physics and partial differential equations in the early twentieth century. Thus, it was seen that numerous physical problems in the theory of equilibria, vibration, quantum theory, etc. could be studied productively via the integral equations that model the phenomena. So it has been, that from the fertile minds of Hilbert, von Neumann, and other giants that the subject of operator theory has grown to a central position in such investigations, and in core mathematics as well. At the heart of this methodology is the deep investigation of the spectrum of an operator and the concommitant study of its invariant subspaces. For the so-called self-adjoint operators, this theory is now a standard technique throughout analysis, and the spectral theorem provides the necessary building blocks for all such operators. Although not quite as complete, spectral theory is significantly well understood for an extensive generalization of the self-adjoint case; viz., the "normal" operators. The current frontier, therefore, in the study of this structure theory rests in the non-normal theory. Professors Conway has long been a leader in the study of subnormal operators, a broad class of operators that arise in applications, and for which there is enough residual normal information to attempt a deeper understanding of their inherent structure. Recent results, suchas the fact that subnormal operators are reflexive and have invariant subspaces, have given an added impetus to the study of their spectral properties. Since many explicit operators that arise in complex function theory, differential geometry, and approximation theory are subnormal, there is a broad range of developing interaction. Professor Conway will study subnormal operators that arise naturally in actions on Hardy spaces, thereby exploring the interaction with function theory. He will also investigate the internal structure theory of subnormal operators, with particular consideration of unitary equivalence, bundle shifts, and the relationship with general operator theory.