Mathematical Sciences: Krein Space Operator theory and Applications

基于算子的因式分解和可拓性质，将发展Krein空间算子理论的一种方法。连续Krein空间算子具有自然的等距和幺正扩展，允许将一般算子的研究简化为更易于处理的情况。指数公式形式的新工具给出了分解存在的条件。寻求了Krein空间交换子提升定理的一种新形式，它具有类似于Hilbert空间情形的应用。一个相关的问题是建立一个统一的Krein空间舒尔类算子值全纯函数的代数和解析理论。这一部分是与几位欧洲同事合作，并以研究不定核和复制核Krein空间为特征的组合。从插值理论和一元函数的例子将被考虑。将Krein空间上自伴随算子的分解性质应用于超不变子空间的研究。其他应用领域包括自伴随算子的紧摄动和在Krein空间中寻找Weyl-von Neumann定理。数值技术将与最近有限矩阵的分解相结合，以努力获得有限维空间数值范围的一般形式的信息。插值问题和线性系统在纯数学和应用数学以及技术中一直很重要。直到最近，数学家才开始研究将其推广到诸如不定内积空间等新情况的可能性。初步结果表明，这条调查路线可能会像经典理论一样富有成果。这些问题自然而然地对不定内积空间上算子理论的其他领域提出了新的问题。这些都与算子的结构性质有关，比如不变子空间和数值范围。与数学相关领域的联系，如经典复分析中的系数问题，也将被探讨。＊＊＊

9501304 Rovnyak An approach to Krein space operator theory will be developed based on factorization and extension properties of operators. Continuous Krein space operators have natural isometric and unitary extensions, allowing a reduction of the study of general operators to a more tractable case. New tools in the form of index formulas give conditions for the existence of factorizations. A new form of the Krein space commutant lifting theorem is sought which has applications similar to the Hilbert space case. A related prmblem is to develop a unified algebraic and analytic theory of the Krein space Schur class of operator-valued holomorphic functions. This part is in collaboration with several European colleagues and features colligations to study indefinite kernels and reproducing kernel Krein spaces. Examples from interpolation theory and univalent functions will be considered. Factorization properties of selfadjoint operators on Krein spaces will be applied to the study of hyperinvariant subspaces. Other areas of application include compact perturbations of selfadjoint operators and a search for a Weyl-von Neumann theorem in Krein spaces. Numerical techniques will be used in combination with recent decompositions for finite matrices in an effort to obtain information on the general form of numerical range for finite-dimensional spaces. Interpolation problems and linear systems have long been important in pure and applied mathematics, as well as in technology. It is only recently that mathematicians have been looking into the possibility of generalizations to new settings such as indefinite inner product spaces. Preliminary results indicate that this line of investigation could turn out to be as fruitful as the classical theory has been. The problems naturally raise new questions about other areas of operator theory on indefinite inner product spaces. These have to do with structural properties of operators such a s invariant subspaces and numerical ranges. Connections with related areas of mathematics such as coefficient problems in classical complex analysis will also be explored. ***