Mathematical Sciences: Problems in Function--Theoretic Operator Theory

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。因此，我们可以看到，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这些现象的积分方程来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，谱定理为所有这些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。一类特别重要的算子被称为“次正规”算子。它们之所以重要，有两个原因。首先，有足够的剩余正常信息来尝试更深入地了解它们的内在结构。最近的研究结果，如次正规算子是自反的，并且具有不变的子空间，为研究它们的谱性质提供了新的动力。其次，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，因此具有广泛的适用性。Kriete教授是亚正规算子的谱理论及其与经典分析，特别是多项式近似的关系的领导者。他的贡献可以追溯到1972年关于切萨罗算子谱理论的里程碑式论文。这项工作为谱理论开辟了新的方向，并对特殊函数理论产生了启示。克里特教授的工作对俄罗斯算子理论学派产生了特别强烈的影响。克里特教授过去几年的工作水平与他早期的工作一样高。在本提案中，Kriete教授将在算子理论和函数理论的界面上研究几个问题，这些问题都表现出极大的兴趣和困难。其中一个问题是确定复变函数理论中经典空间上的复合算子何时具有次正规伴随。第二个问题涉及到收缩操作符的功能模型。他还将继续他在经典多项式近似中的应用，重点是分裂问题。

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, vibrations, 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 for their inherent structure. Recent results, such as the fact that subnornmal 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 Kriete is a leader in the spectral theory of subnormal operators and its relationship to classical analysis, especially polynomial approximation. His contributions date back to a landmark paper on the spectral theory of the Cesaro operator in 1972. This work created new directions in spectral theory, with implications for the theory of special functions. Professor Kriete's work has had a particularly strong influence on the Russian school of operator theory. Professor Kriete's work of the past few years has been of the same high calibre as his earlier work. In the present proposal, Professor Kriete will investigate several problems at the interface of operator theory and function theory, all of which exhibit both significanct interest and difficulty. One such problem is the determination of when a composition operator on classical spaces in complex function theory has a subnormal adjoint. A second involves functional models for contraction operators. He will also continue his applications to classical polynomial approximation with emphasis on questions of splitting.