Mathematical Sciences: Problems in Operator Theory and Nonselfadjoint Operator Algebras

这个项目是关于算子论的数学研究。这一理论的最初推动力来自于量子力学中的一个基本见解，即通常被视为标量的物理量可以更好地表示为希尔伯特空间上的算子，在那里它们可以以更有趣的方式相互作用。自那以后，对运算符及其如何转换底层空间的分析已经获得了除物理学之外的独特存在。更具体地说，Bercovici教授将研究一类新的非自伴算子代数，称为对偶代数，重点是不变子空间的存在和分类。他将丰富n元自伴算子的扰动理论，还将研究斜Toeplitz算子。后者的研究对工程上的控制理论问题有着重要的影响。

This project is for mathematical research in operator theory. The original impetus for this theory came from a fundamental insight in quantum mechanics, namely that physical quantities normally regarded as scalars are better represented by thinking of them as operators on Hilbert space, where they can interact in a more interesting fashion. The analysis of operators and how they transform the underlying space has since acquired a distinguished existence of its own, apart from physics. More specifically, Professor Bercovici will study a new class of nonselfadjoint operator algebras called dual algebras, with an emphasis on the existence and classification of invariant subspaces. He will enrich the perturbation theory of n-tuples of selfadjoint operators, and will also investigate skew Toeplitz operators. The latter study impinges significantly on control theory issues in engineering.