Mathematical Sciences: Structure of Toeplitz and CompositionOperators

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。一类特别重要的这类算子被称为“次正常”。它们之所以意义重大，有两个原因。首先，有足够的剩余正常信息试图更深入地了解它们的内在结构。最近的结果，例如次正规算子是自反的，并且有不变子空间的事实，增加了对它们的谱性质的研究。其次，由于复变函数论、微分几何和逼近理论中出现的许多显式算子都是非正规的，因此具有广泛的适用性。考恩教授是次正常算符的谱理论及其与经典分析关系的领军人物。他的程序涉及对单位圆盘的Hardy空间上的显式算子的研究，以便深入了解更一般的问题。特别是，他建议继续他对复合算子、Toeplitz算子和相关算子的研究。在最近的协作工作中，他发现了一类不正常的合成算子。考恩教授建议扩展这一类别，并进一步研究非正态现象。比次正规更一般的算子出现在Krein空间上的乘法算子上。考恩教授建议启动对这类运算符的研究。在早期的工作中，他通过构造次正规Toeplitz算子解决了一个老问题，这些算子既不正规也不解析。考恩教授建议研究更普遍的亚正常Toeplitz算子，目的是找到决定亚正常的符号的条件。

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. A particularly important class of such operators are referred to as "subnormal". They are significant for two reasons. First, there is enough residual normal information to attempt a deeper understanding of their inherent structure. Recent results, such as the fact that subnormal operators are reflexive and have invariant subspaces, have given an added impetus to the study of their spectral properties. Secondly, since many explicit operators that arise in complex function theory, differential geometry, and approximation theory are subnormal, there is a broad range of applicability. Professor Cowen is a leader in the spectral theory of subnormal operators and its relationship to classical analysis. His program involves the investigation of explicit operators on the Hardy space of the unit disk in order to gain insight into more general problems. In particular, he proposes to continue his investigation of composition operators, Toeplitz operators, and related operators. In recent collaborative work he found a class of composition operators which are subnormal. Professor Cowen proposes to extend this class, and to further investigate subnormality. Operators more general than subnormal arise as multiplication operators on Krein space. Professor Cowen proposes to initiate the study of such operators. In earlier work, he solved an old problem by constructing subnormal Toeplitz operators which are neither normal nor analytic. Professor Cowen proposes to investigate the more prevalent hyponormal Toeplitz operators with the goal of finding conditions on the symbol that determine hyponormality.