Mathematical Sciences: Problems in Function-Theoretic Operator Theory

Kriete教授将研究算子论和函数论卓有成效地相互作用的四个问题。第一个问题是完全确定与Cesaro算子相关的特定循环次正规算子的不变子空间格。第二个是关于作用在Bergman空间上的复合算子。第三个问题是加权多项式逼近中的一个经典问题，它是研究次正规算子的一个有趣的问题。最后是关于压缩算子的de Brange-Rovnyak泛函模型中范数的极值行为。这里设想的研究是关于解析函数的，这是一个多世纪以来数学的核心问题。它们可以被不同地定义为某一简单偏微分方程组的解，或将平面区域映射到其他平面区域的映射，其角度保持不变，除非在孤立的奇点处，或者定义为适当精确意义上的多项式的极限。与描述它们的不同方式相对应，解析函数可以用不同的方式来研究。这个项目中所采用的方法是算子论，通过以特定的方式获取多项式及其极限来建立函数的希尔伯特空间，然后研究由函数自然产生的空间上的算子，例如通过乘法或通过合成。相反，这些函数论算子在算子理论中往往是非常普遍的，所以通过研究这些函数，人们可以从抽象的角度了解大量关于算子的知识。Kriete教授的项目是在这条双向街道上穿行。

Professor Kriete will investigate four problems in which operator theory and function theory interact in a fruitful way. The first of these is to determine completely the invariant subspace lattice of a particular cyclic subnormal operator related to the Cesaro operator. The second concerns composition operators acting on Bergman spaces. The third is a classical question in weighted polynomial approximation which is of interest in the study of subnormal operators. The last is about the extremal behavior of the norms in the de Branges - Rovnyak functional model for contraction operators. The research envisioned here is about analytic functions, a central preoccupation of mathematics for well over a century. They can be defined variously as solutions of a certain simple system of partial differential equations, as maps which take planar regions to other planar regions in a way that preserves angles except at isolated singularities, or as limits of polynomials in a suitably precise sense. Corresponding to the diverse ways of describing them, analytic functions can be studied in a variety of ways. The approach favored in this project is operator-theoretic, building up Hilbert spaces of functions by taking polynomials and their limits in a particular fashion, then studying operators on the spaces that arise naturally from the functions, e.g. by multiplication or by composition. Conversely, these function-theoretic operators often turn out to be quite general in operator theory, so by studying the functions one can learn a great deal about operators in the abstract. Professor Kriete's project is to travel this two-way street in both directions.