Mathematical Sciences: Collaboration in Matrix Analysis

该方案的研究涉及到矩阵和算子的理论。在算子理论中，关注的将是算子值函数，而中心主题将是因式分解问题。其中一些主题包括各种类型的算子函数的谱数据的分析，给定其谱数据的一类算子函数的综合，自伴算子多项式的特殊因式分解，具有紧谱的解析算子函数的因式分解与其线性化的不变子空间之间的对应，以及亚纯算子函数的局部和整体等价。在矩阵理论中，这些问题包括矩阵补全问题，定性矩阵理论中的谱分布问题，无限维谱分配问题，保持可控性的线性变换，特征多项式的推广，以及子空间的解析族。总的目标将是了解统一的结构，该结构可能支持每个领域的其他问题解决方案。

The research in this proposal is concerned with the theory of matrices and operators. In operator theory the concern will be in operator valued functions and the central theme will be factorization problems. Some of the topics are analysis of spectral data of various classes of operator functions, synthesis of an operator function in a given class given its spectral data, special factorizations of self adjoint operator polynomials, correspondence between factorizations of an analytic operator function with compact spectrum and invariant subspaces of its linearization and local and global equivalence of meromorphic operator functions. In matrix theory the problems will include matrix completion problems, spectral distribution in qualitative matrix theory, infinite dimensional spectral assignment problems, linear transformations that preserve controllability, a generalization of the characteristic polynomial, and analytic families of subspaces. A general goal will be the understanding of the unifying structure which may support additional problem solutions in each area.