Problems in Modern Analysis

申请者将研究算子理论问题，不变量子空间和稳定性相关问题。提出了一种利用不同类型的极值向量构造不变子空间的新方法。到目前为止，这已经导致了大型算子类的不变子空间的新的和统一的构造，包括与紧算子或正规算子交换的算子。它还导致了所谓的双序列定理的增强，它与不变子空间相连。有几个方向，以完善和改进这一技术和提议者打算这样做。对于算子和向量，在环性、超环性或超环性扰动下的稳定性研究仍处于起步阶段。这与研究哪些特性被广泛接受密切相关。小类操作符。提议者打算进一步进行这些研究。算子及其不变子空间的研究可以看作是线性代数的推广，其中研究矩阵及其特征向量。虽然这些都是“纯数学”中的问题，但它们与线性代数在物理学、生物学、遗传学、经济学和其他科学以及工业中的许多应用密切相关。算子可以看作是无限矩阵，是研究依赖无界参数数量的大型复杂系统的合适工具。

The proposer will work on Operator Theory problems,problems related to invariant subspaces and tostability. The proposer has developed a new techniqueto construct invariant subspaces by using differentkinds of extremal vectors. This has, so far, led tonew and unified constructions of invariant subspaces for large classes of operators, including operatorscommuting with compact or normal operators. It has also led to strengthenings of so called Two SequencesTheorems, which are connected to invariant subspaces.There are several directions to refine and improve thistechniques and the proposer intends to do so. The study of stability under perturbations of cyclicity,supercyclicity or hypercyclicity is still in itsbeginning, both for operators and vectors. This isstrongly connected to the study of which properties arecarried by large resp. small classes of operators. Theproposer intends to carry these studies further.The study of operators and their invariant subspaces canbe seen as a generalization of linear algebra, where matrices and their eigenvectors are studied. Althoughthese are problems in "pure" mathematics, they areclosely connected to the many applications of linearalgebra to Physics, Biology, Genetics, Economics and other sciences as well as to the applications in industry. Operators can be seen as infinite matrices and are a suitabletool for the study of large and complex systems depending on anunbounded number of parameters.