Mathematical Sciences: Structure of Toeplitz and CompositionOperators

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。 为了所谓的 自伴算子，这个理论现在是一个标准的技术 在整个分析中，谱定理提供了 为所有这些运营商提供必要的构建模块。 尽管未 同样完整的是，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，例如低于正常的事实， 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 是低于正常的，有一个广泛的适用性。 考恩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 他的程序涉及到对 单位圆盘的哈代空间，以便深入了解 更普遍的问题。 特别是，他建议继续 他对复合算子，Toeplitz算子， 和相关运营商。 在最近的合作工作中，他发现 一类次正规的复合算子。 教授 Cowen建议扩展这个类，并进一步研究 低于正常水平。 比次正规算子更一般的算子出现为 Krein空间上的乘法算子 柯文 建议启动对此类运营商的研究。 在更早 工作，他解决了一个老问题，通过构建亚正常Toeplitz 既不是正规的也不是解析的算子。 柯文 建议调查更普遍的亚正常Toeplitz 运算符的目标是在符号上找到条件， 确定亚常态。

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.