Mathematical Sciences: Research on Operators in Hilbert Space

本项目是算子理论中的数学研究， 这门学科起源于量子力学（其中 希尔伯特空间上的算子被用来代替标量， 表示物理量），但它早已获得 一个充满活力的生命。 操作员之间相互作用 比标量和单个运算符更丰富 携带大量的结构（频谱、数值不变量 例如索引等），其说明是永久的 挑战主题。 Fialkow教授的部分研究涉及光谱， 指数为所谓的基本运营商，这是线性的 左乘法和右乘法的组合。 主旨 这里是恢复有关给定的基本信息， 操作员在多变量光谱数据方面， 它是由什么组成的。 另一个方面与 拟相似性，两个算子是拟相似的，如果它们是 通过一对内射变换交织在一起 具有密集范围;目标是表征操作员 这与单边转移是类似的。

This project is mathematical research in operator theory, a subject that had its origins in quantum mechanics (where operators on Hilbert space are used in place of scalars to represent physical quantities), but that has long since acquired a vigorous life of its own. Operators interact with one another in a much richer way than do scalars, and individual operators carry a great deal of structure (spectrum, numerical invariants such as index, and so forth) whose elucidation is a perpetual challenge to the subject. Part of Professor Fialkow's research deals with spectrum and index for so-called elementary operators, which are linear combinations of left and right multiplications. The main thrust here is to recover information about the given elementary operator in terms of multivariable spectral data from the pieces of which it is composed. Another aspect has to do with quasisimilarity, two operators being quasisimilar if they are intertwined left and right by a pair of injective transformations with dense range; the objective is to characterize the operators that are quasimilar to the unilateral shift.