Mathematical Sciences: Two-Isometries and Dirichlet-Type Spaces

Richter教授的项目将研究希尔伯特空间上一类称为二等距的算子，它满足定义等距的保范条件的一个较弱的二阶版本。二等距的一个重要例子是在函数空间上乘以z的算子，这些函数解析地将圆盘映射到一个有限面积的区域，即经典狄利克雷空间上的经典狄利克雷位移。本研究的实质是利用算子理论的结果来研究Dirichlet型空间中的函数，反过来利用函数理论来构造抽象二等距的模型。一个中心问题是狄利克雷位移的不变子空间格及其推广的刻画。这里设想的数学研究与圆盘上的解析函数有关。这些可以从几何上描述为圆盘到平面的连续映射，这些映射保持了角度，除了分散的奇点。解析函数在一个多世纪以来一直在数学中扮演着重要的角色，这反映了它们也可以用许多其他的方式来描述。研究解析函数的一种方法是将它们分组到线性空间中，最有效的是由适当的有限条件定义的希尔伯特空间，并考虑在这些空间中自然出现的某些算子的行为。其中一个空间是由圆盘上所有解析函数组成的狄利克雷空间，这些解析函数将圆盘映射到一个面积有限的平面区域。这个空间中函数的乘法算子（及其推广）将由Richter教授研究和描述。

Professor Richter's project will investigate a class of operators on Hilbert space called two-isometries, which satisfy a weaker, order two version of the norm-preserving condition that defines isometries. An important example of a two-isometry is the operator of multiplication by z on the space of functions that map the disc analytically to a region with finite area, i.e. the classical Dirichlet shift on the classical Dirichlet space. The essence of the projected research is to use operator-theoretic results to study functions in spaces of Dirichlet type, and conversely to use function theory to construct models for abstract two-isometries. A central problem is that of characterizing the invariant subspace lattice of the Dirichlet shift and its generalizations. The mathematical research envisioned here has to do with analytic functions on the disc. These may be described geometrically as the continuous mappings of the disc into the plane that preserve angles, except perhaps at scattered singular points. It is a reflection of the important role analytic functions have played in mathematics for well over a century that they can be described in numerous other ways as well. One method of studying analytic functions is to group them into linear spaces, most fruitfully Hilbert spaces defined by an appropriate finiteness condition, and consider the behavior of certain operators that arise naturally on these spaces. One such space is the Dirichlet space consisting of all analytic functions on the disc which map the disc to a planar region with finite area. The operator of multiplication of functions in this space (and its generalizations) by the independent variable will be studied and characterized by Professor Richter.