Mathematical Sciences: Subnormal Operators and Related Topics

算子理论是现代分析的核心学科。它起源于20世纪初对数学物理和偏微分方程的研究。由此可见，平衡理论、振动理论、量子理论等许多物理问题都可以通过模拟现象的积分方程进行富有成效的研究。因此，从希尔伯特、冯·诺伊曼和其他巨人的丰富思想中，算符理论这门学科已经发展到这些研究的中心位置，在核心数学中也是如此。该方法的核心是对算子谱的深入研究及其不变子空间的伴随研究。对于所谓的自伴随算子，这个理论现在是整个分析的标准技术，谱定理为所有这样的算子提供了必要的构建块。虽然不完全完整，但谱理论对于自伴随情况的广泛推广是很好的理解；即“正常”操作符。因此，目前研究这一结构理论的前沿在于非正态理论。长期以来，Conway教授一直是亚正规算子研究的领导者，亚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理解它们的内在结构。最近的结果，如亚正规算子是自反的，并且具有不变的子空间，为研究它们的谱性质提供了额外的动力。由于在复函数理论、微分几何和近似理论中出现的许多显式算子是次正规的，因此有广泛的发展相互作用。Conway教授将研究在Hardy空间上的作用中自然产生的亚正规算子，从而探索与函数理论的相互作用。他还将研究亚正规算子的内部结构理论，特别考虑幺正等价、束位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.