Mathematical Sciences: Problems in Function--Theoretic Operator Theory

算子理论是现代分析中的一个中心学科。 它的起源在于研究数学物理和部分 微分方程在世纪早期。 因此 我们发现，在量子力学理论中， 平衡、振动、量子理论等都可以研究 有效地通过模拟现象的积分方程。 因此，从希尔伯特丰富的思想中， 诺依曼和其他巨人认为算子理论的主题 在这样的调查中成长为中心地位，在核心 数学也是。 这种方法的核心是 对算子谱的深入研究， 不变子空间的伴随研究。 对于如此- 称为自伴算子，这个理论现在是一个标准 技术贯穿分析，谱定理提供了 所有这些运营商的必要基石。 虽然 虽然不那么完整，但光谱理论 理解为自伴的广泛推广 情况;即，“正常”的运营商。 目前的前沿， 因此，在对这种结构理论的研究中， 非常态理论 其中一类特别重要的 操作者被称为“次正常”。 它们意义重大 有两个原因 首先，有足够的剩余正常 试图更深入地了解其固有的信息， 结构 最近的结果，例如， 算子是自反的，具有不变子空间，给出了 这对研究它们的光谱特性是一个额外的推动力。 其次，由于许多显式运算符出现在复杂的 函数论、微分几何和逼近理论 虽然低于正常水平，但适用范围广泛。 Kriete教授是光谱理论的领导者， 次正规算子及其与经典分析的关系， 尤其是多项式逼近。 他的贡献可以追溯到 一篇关于塞萨罗光谱理论的里程碑式论文 1972年的运营商。 这项工作开创了光谱学的新方向。 理论，与特殊功能的理论的影响。 Kriete教授的研究对 关于俄罗斯算子理论学派 Kriete教授 过去几年的工作一直是同样的高素质， 他早期的工作。 在目前的建议中，Kriete教授将 研究算子理论界面上的几个问题 和函数论，都表现出既有意义， 兴趣和困难。 其中一个问题是 当复空间中的经典空间上的复合算子 函数论有一个次正规伴随。 第二个涉及到 收缩算子的函数模型。 他还将 继续他对经典多项式逼近的应用 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.