Mathematical Sciences: Operators on Spaces of Analytic Functions

DMS-9622914 PI：里士满威廉·罗斯大学首席研究员将对解析函数空间上的算子进行研究。这些问题探讨了泛函分析与复变函数论之间的关系。该方案的第一部分研究了解析函数的一般Hilbert空间上的“后向移位算子”的不变子空间。在提案的第二部分，主要研究人员计划检查“新月”区域上的Bergman空间，并检查在线性变换“乘以z”下不变的某些子空间。在提案的第三部分，首席调查员计划研究“标准新月”(由两个内部相切的圆界定的区域)上多项式的闭合。如前所述，这些类型的问题结合了两个数学领域，泛函分析和复函数理论。这种联系是由Arne Beurling于1949年在他关于移位运算符的研讨会论文中首次注意到的。科伦布卢姆、尼科尔斯基、哈文、希罗科夫和沙莫扬等几位俄罗斯人研究了这一转变的进一步结果。在这项建议中，我们将研究这些相同的问题，但将更密切地关注所涉及的几何问题。

DMS-9622914 PI: William Ross University of Richmond The principal investigator will conduct research on operators on spaces of analytic functions. Such problems explore the relationship between functional analysis and complex function theory. The first part of the proposal examines the invariant subspaces of the "backward shift operator" on a general Hilbert space of analytic functions. In the second part of the proposal, the principal investigator plans to examine the Bergman space on a "crescent" domain and examine certain subspaces which are invariant under the linear transformation "multiplication by z". In the third part of the proposal, the principal investigator plans to look at the closure of the polynomials on the "standard crescent" (the region bounded by two internally tangent circles). As mentioned earlier, these types of problems combine two fields of mathematics, functional analysis and complex function theory. This link was first noticed by Arne Beurling in 1949 in his seminar paper on the shift operator. Further results on the shift were taken up by several Russians such as Korenblum, Nikolskii, Havin, Shirokov, and Shamoyan. In this proposal we examine these same problems but will pay closer attention to the geometry involved.