Positivity Conditions in Complex Analysis

提案编号：复分析的一个重要方面涉及全纯函数的绝对值不等式，或者更一般地说，涉及全纯映射的范数不等式。在最近的工作中，proposerhas开发了一个系统的方法来解决这个问题。他给出了一个复变模拟希尔伯特的第17问题，适用于适当的holomorphicmapping技术，并（与卡特林）证明了等距imbeddingtheorem全纯向量丛。此外，他最近写了卡鲁斯专题题为“不平等的复杂性分析”，从而奠定了基础，为更多的工作在这一领域。目前的提案描述了如何扩展这些研究方向，并讨论了可能的数学应用。虽然提案侧重于纯数学，但人们可以想象其中一些工作在科学上的应用。例如，在量子力学中，可观测量是希尔伯特空间上的自伴（厄米特）算子。该建议允许一个更一般的方法，通过描述一个概念的自伴非线性映射。提议者工作的主要思想之一是描述此类映射的各种积极性概念。出现了几种新情况;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。特别是提议者希望开发应用程序的非线性模拟的柯西-施瓦茨不等式。

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.