Mathematical Sciences: Krein Space Operator theory and Applications

小行星9501304 一种方法克莱因空间算子理论将发展 基于算子的因子分解和扩张性质。 连续Krein空间算子具有自然等距性， 酉扩展，允许减少一般的研究 操作员到一个更容易处理的情况下。新的工具， 指数公式给出了 分解Krein空间交换子提升的一种新形式 定理寻求具有类似于希尔伯特的应用 太空箱一个相关的问题是发展一个统一的代数 Krein空间Schur类的解析理论 算子值全纯函数这部分是在 与几位欧洲同事合作， 研究不定核和再生核的结合 Krein空间 插值理论和单叶函数的例子 功能将被考虑。的因式分解性质 Krein空间上的自伴算子将被应用于 超不变子空间的研究其他应用领域 包括自伴算子紧扰动和 在Krein空间中寻找Weyl-von Neumann定理。数值 技术将与最近的分解相结合， 有限矩阵，以获得有关 有限维空间数值域的一般形式。 插值问题和线性系统一直是 在纯数学和应用数学中很重要， 技术. 直到最近，数学家们才被 研究推广到新环境的可能性 例如不定内积空间。 初步结果 表明这条调查路线可能会被证明是 正如经典理论一样富有成果。 的问题 自然会对运营商的其他领域提出新的问题 不定内积空间理论 这些都与 s不变子空间等算子的结构性质 和数值范围。 与相关领域的联系 数学，如经典复形中的系数问题 还将进行分析。 ***

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. ***