Mathematical Sciences: Research on Operators in Hilbert Space

这个项目是对算子理论的数学研究，这一学科起源于量子力学（希尔伯特空间中的算子被用来代替标量来表示物理量），但它早已获得了自己的蓬勃发展。算符以比标量更丰富的方式相互作用，并且单个算符携带大量的结构（谱，数值不变量，如索引，等等），其阐明是对该主题的永恒挑战。Fialkow教授的部分研究涉及所谓的初等算符的谱和索引，它是左乘和右乘的线性组合。这里的主要推力是根据多变量光谱数据从组成它的碎片中恢复有关给定初等算子的信息。另一方面与拟相似有关，如果两个算子被一对范围密集的内射变换左右缠绕在一起，它们就是拟相似的；目的是表征与单边位移拟相似的算子。

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.