Positivity Conditions in Complex Analysis

复分析的一个重要方面是关于全纯函数的绝对值不等式，或者更一般地说，关于全纯映射的范数不等式。在最近的工作中，该提倡者对这一主题提出了一种系统的方法。他给出了希尔伯特第17问题的复变量模拟，将该技术应用于适当的全纯映射，并（与卡特林）证明了全纯向量束的等距嵌入定理。他最近还撰写了Carus专著《来自复杂分析的不等式》，从而为该领域的更多工作奠定了基础。目前的建议描述了如何扩展这些研究方向，并讨论了可能的数学应用。尽管该提案侧重于纯数学，但人们可以想象其中一些工作将应用于科学。例如，在量子力学中，一个可观测值是希尔伯特空间上的自伴随算子（厄米算子）。通过描述非线性映射的自伴随性概念，该建议允许更一般的方法。提议者工作的主要思想之一是描述这种映射的各种正性概念。出现了几个新情况；这些条件在量子力学的线性情况下是等价的，但在一般情况下是不同的，因此可能导致在物理学中的应用。特别地，提议者希望开发Cauchy-Schwarz不等式的非线性模拟的应用。

Proposal Number: DMS-0200551PI: John P. D'AngeloABSTRACTAn important aspect of complex analysis concerns inequalitieson absolute values of holomorphic functions or, more generally,on norms of holomorphic mappings. In recent work the proposerhas developed a systematic approach to this subject. Hehas given a complex variables analogue of Hilbert's 17thproblem, applied the techniques to proper holomorphicmappings, and (with Catlin) proved an isometric imbeddingtheorem for holomorphic vector bundles. Also he has recentlywritten a Carus Monograph entitled "Inequalities from ComplexAnalysis", thereby laying a foundation for more work in thisarea. The current proposal describes how to expand theseresearch directions and discusses possible mathematicalapplications.Although the proposal focuses on pure mathematics, one canimagine the application to the sciences of some of this work.For example, in Quantum Mechanics, an observable is aself-adjoint (Hermitian) operator on a Hilbert space. Theproposal allows for a more general approach, by describinga notion of self-adjointness for non-linear mappings. One ofthe main ideas of the proposer's work has been to describevarious notions of positivity for such mappings. Several newconditions arise; these conditions are equivalent in thelinear case from Quantum Mechanics, but distinct in general,and therefore may lead to applications in physics. Inparticular the proposer hopes to develop applicationsof the non-linear analogue of the Cauchy-Schwarz inequality.