Mathematical Sciences: Operator Theory and Analysis

Rovnyak教授将继续研究复杂分析中比伯巴赫猜想证明的后续发展。后者导致了在包含和推广狄利克雷空间的某些Krein空间中的压缩替代变换的研究。这些变换的性质将从不变子空间理论和不定度量空间上的线性算子理论的角度来探讨。一个长期目标是一个系数体问题，它寻求表征幂级数的初始段，幂级数表示将磁盘映射到自身的归一化黎曼映射函数。本文还研究了Krein空间上缩算符的相关抽象可拓问题和交换子提升定理。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中心关注点。它们可以被定义为某种简单的偏微分方程组的解，可以被定义为将平面区域映射到其他平面区域的映射，除了在孤立的奇点处保持角度，也可以被定义为多项式在适当精确意义上的极限。与描述解析函数的不同方式相对应，解析函数可以用多种方式进行研究。在这个项目中青睐的方法是算子理论，通过以特定的方式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，例如通过乘法或复合。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，人们可以学到很多抽象的算子。

Professor Rovnyak will pursue developments consequent upon the proof of the Bieberbach conjecture in complex analysis. The latter has led to a study of contractive substitution trans- formations in certain Krein spaces that include and generalize the Dirichlet space. Properties of these transformations will be explored both from the point of view of invariant subspace theory and the theory of linear operators on spaces with indefinite metric. A long-range goal is a coefficient body problem that seeks to characterize initial segments of power series which represent normalized Riemann mapping functions that map the disk into itself. Related abstract extension problems and commutant lifting theorems for contraction operators on Krein spaces will also be studied. 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.