Applications of Krein space operator theory

翻译后摘要：该建议涉及算子理论和功能theoryinvolving广义Schur和Nevanlinna类和两个等距庞特里亚金空间。 本文将研究插值理论中Nudelman问题的一个推广。 在经典情形下，M. Rosenblum和PI使用交换子提升定理。 该准则包含了一系列经典的插值结果，而交换提升定理的Ball和Helton推广将Nudelman问题推广到了具有类似应用的广义Schur类。 边界插值和希尔伯特变换的新问题出现在不定的情况下，并将研究在联合工作与T。Constantinescu和A. Dijksma.D. Z. Arov和PI将研究单位圆盘上J-压缩矩阵值函数的达林顿综合。 现在的状态空间是庞特里亚金空间。 该方法遵循以前的工作Arovand还使用了分解理论的伪continuablefunctions罗森布卢姆和PI。 L. A. Sakhnovich和PI将用算子恒等式方法研究正则微分和差分方程正问题和反问题的不定情形。 刚拿到博士学位。C的论文。Hellings在Pontryagin空间上推广了Richter关于解析二等距的定理。 这一领域是新的，包括一些开放的问题和进一步发展的可能性。该提案涉及的问题，在数学领域，有传统的连接与工程和控制理论，特别是设计稳定的反馈控制器和passivenetworks。 插值理论的主要问题是从函数在指定点的给定值等部分信息重构函数。 通常，问题的数据满足正性条件取决于所考虑的函数类。 该方案将研究一种统一的方法来处理一类这样的问题，其中的正性条件被较弱的假设所取代。 谱理论的逆问题要求从观测到的谱性质重构一个算子，它与插值理论有关，削弱正性的影响将再次被考虑。 其他问题涉及到一个被动系统嵌入到一个conservativesktop：扩展将被认为是在系统中包含主动和被动元素。 保守系统在某种意义上对应于等距算子，另一组问题探索了在没有正性的情况下等距算子的推广。 该项目的资金还将为弗吉尼亚大学数学系的一个本科生研究项目提供支持，该项目将学生与数学教授一对一地进行为期约8周的项目。

Abstract:The proposal is concerned with operator theory and function theoryinvolving the generalized Schur and Nevanlinna classes andtwo-isometries on Pontryagin spaces. A generalization of Nudelman'sproblem in interpolation theory will be studied. In the classicalcase, existence criteria were given by M. Rosenblum and the PI usingthe commutant lifting theorem. The criteria imply a series ofclassical interpolation results with both interior and boundary data.The Ball and Helton generalization of the commutant lifting theoremyields an extension of Nudelman's problem to the generalized Schurclass which has similar applications. New problems for boundaryinterpolation and Hilbert transforms arise in the indefinite case andwill be studied in joint work with T. Constantinescu and A. Dijksma.D. Z. Arov and the PI will study Darlington synthesis forJ-contractive matrix-valued functions in the unit disk. State spacesnow are Pontryagin spaces. The approach follows previous work by Arovand also uses the factorization theory for pseudo-continuablefunctions by Rosenblum and the PI. L. A. Sakhnovich and the PI willstudy indefinite cases of direct and inverse problems for canonicaldifferential and difference equations by the method of operatoridentities. A recent Ph.D. dissertation by C. Hellings obtains ageneralization of Richter's theorem on analytic two-isometries onPontryagin spaces. This area is new and includes a number of openquestions and possibilities for further developments.The proposal is concerned with problems in areas of mathematics whichhave traditional connections with engineering and control theory andespecially the design of stabilizing feedback controllers and passivenetworks. The main problem of interpolation theory is to reconstructa function from partial information such as given values of thefunction at specified points. Usually the data of a problem satisfy apositivity condition depending on the class of functions underconsideration. The proposal will study a unified method to deal withone class of such problems in which the positivity conditions arereplaced by weaker assumptions. The inverse problem of spectraltheory, which asks to reconstruct an operator from observed spectralproperties, is related to interpolation theory, and the effect ofweakening positivity will again be considered. Other problems arerelated to the embedding of a passive system into a conservativesystem: extensions will be considered in which systems contain activeas well as passive elements. Conservative systems correspond in somesense to isometric operators, and another group of problems exploresgeneralizations of isometric operators in the absence of positivity.The projects are in collaboration with colleagues in the U.S. andEurope. Funding of the project will also provide support for anundergraduate research program in the Mathematics Department of theUniversity of Virginia that matches students one-on-one withmathematics professors for projects of approximately eight weeks in