Mathematical Sciences: Two-Isometries and Dirichlet-Type Spaces

里希特教授的项目将研究一类 希尔伯特空间上的算子称为两等距，满足一个 更弱的，二阶保范条件的版本， 定义等距。两等距的一个重要例子是 在函数空间上乘以z的算子， 将盘解析地映射到具有有限面积的区域，即， 经典Dirichlet空间上的经典Dirichlet移位。的 该项目研究的本质是使用运营商理论 研究Dirichlet型空间中函数的结果，以及 相反，使用函数理论来构建模型， 抽象二等距一个核心问题是， 刻画Dirichlet的不变子空间格 转移及其推广。 这里设想的数学研究与 光盘上的解析函数。这些可被描述 在几何上，作为圆盘到 保持角度的平面，除了可能在分散的奇异 点它反映了分析的重要作用 函数在数学中的作用已经存在了世纪， 它们也可以用许多其他方式来描述。一种方法 研究解析函数的一个重要方法是把它们分成线性的 空间，最富有成效的希尔伯特空间定义了一个适当的 有限性条件，并考虑某些行为 在这些空间中自然产生的算子。一个这样的空间是 Dirichlet空间由所有解析函数组成， 盘，其将盘映射到具有有限面积的平面区域。的 在这个空间中函数的乘法运算符（及其 一般化）的独立变量将进行研究， 由Richter教授描述。

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.