Positivity Conditions in Complex Analysis

建议编号：DMS-0200551PI：John P.D‘Angelo Abstrant复分析的一个重要方面涉及全纯函数的绝对值的不等式，或者更一般地，关于全纯映射的范数的不等式。在最近的工作中，这位提出者对这一主题发展了一种系统的方法。他给出了Hilbert第17个问题的复变量类比，将这些技巧应用于真全纯映射，并(与Catlin)证明了全纯向量丛的一个等距嵌入定理。他最近还写了一本CARU专著，题为《复分析中的不平等》，从而为这一领域的更多工作奠定了基础。目前的方案描述了如何扩展研究方向，并讨论了可能的数学应用。虽然方案集中在纯数学上，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，一个可观的是希尔伯特空间上的一个自伴(厄米)算符。该提议允许更一般的方法，通过描述非线性映射的自伴性的概念。这位提出者工作的主要思想之一就是描述了这类映射的各种正性概念。出现了几个新的条件；这些条件在量子力学的线性情况下是等价的，但总体上是不同的，因此可能导致在物理学中的应用。特别是，提出者希望发展柯西-施瓦茨不等式的非线性类比的应用。

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.