Commutative Algebraic Methods in Operator Theory

该项目的目的是发展一个Fredholm理论，包括一个单一的运营商和一个元组的交换运营商作用于一个共同的希尔伯特空间。 Koszul复杂的方法将被强调，以介绍交换代数的思想和方法，算子理论。该策略是引入稳定的不变量，如塞缪尔多重算子理论计算多变量Fredholm指数。这有许多连接与现有的主题，如余维不变子空间的伯格曼空间和狄利克雷空间，Toeplitz运营商的哈代空间，Arveson的曲率不变的对称福克空间，函数理论的哈代空间在高维域，Apostol的三角表示。Fredholm指数的线性算子是一个最激烈的研究数值不变量在数学。著名的Atiyah-Singer指标定理是数学中最深刻的结果之一，其内容是关于如何计算一类特殊算子的Fredholm指标。这个项目将侧重于开发一个多变量版本的传统的一个变量Fredholm理论。这使人们能够引入思想从交换代数，一个看似遥远的领域在数学，到多变量算子理论，这反过来又揭示了新的光传统问题的一个变量算子理论。

The project aims at developing a Fredholm theory encompassing both a single operator and a tuple of commuting operators acting on a common Hilbert space. The Koszul complex approach will be emphasized in order to introduce ideas and methods in commutative algebra to operator theory. The strategy is to introduce stabilized invariants such as the Samuel multiplicities into operator theory to calculate the multivariable Fredholm index. This has many connections with existing topics such as the codimension of invariant subspaces of the Bergman space and the Dirichlet space, Toeplitz operators on the Hardy space, Arveson's curvature invariant on the symmetric Fock space, function theory on Hardy spaces over higher dimensional domains, Apostol's triangular representation.The Fredholm index of a linear operator is one of the most intensely studied numerical invariants in mathematics. The content of the celebrated Atiyah-Singer Index Theorem, one of the deepest results in mathematics, is on how to calculate the Fredholm index of a special class of operators. This project will focus on developing a multivariable version of the traditional one variable Fredholm theory. This allows one to introduce ideas from commutative algebra, a seemingly distant area in mathematics, into multivariable operator theory, which in turn shed new light on traditional problems in one variable operator theory.