Mathematical Sciences: Multivariable Spectral Theory and Complex Analysis

普京将调查两组问题。第一部分涉及多变量Bergman空间上的Toeplitz算子及其与若干复变量函数理论的相关性。他将这类算子的谱问题与具有l2生长控制的伪凸域上的函数理论联系起来。第二组问题与欧几里得空间矩理论有关。Putinar将尝试使用特定的算子理论来确定解决半代数子集上某些矩问题的方法。算符理论是研究矩阵的无限维推广的数学部分。特别地，当限制在有限维子空间中时，算子具有通常的线性性质，因此可以用矩阵表示。算子理论的中心问题是对满足附加条件的算子进行分类，这些附加条件是根据相关算子（如伴随算子）或根据基础空间给出的。算子理论是许多数学的基础，也是数学在其他科学中的许多应用的基础。

Putinar will investigate two sets of problems. The first concerns Toeplitz operators on multivariable Bergman spaces and their relevance for the function theory of several complex variables. He will relate the spectral problems for this class of operators to the theory of functions in pseudoconvex domains with L2-growth control at the boundary. The second set of problems is related to the theory of moments in Euclidean n- space. Putinar will try to determine methods for solving certain moment problems on semialgebraic subsets by using specific operator theories. Operator theory is that part of mathematics that studies the infinite dimensional generalizations of matrices. In particular, when restricted to finite dimensional subspaces, an operator has the usual linear properties, and thus can be represented by a matrix. The central problem in operator theory is to classify operators satisfying additional conditions given in terms of associated operators (e.g. the adjoint) or in terms of the underlying space. Operator theory underlies much of mathematics, and many of the applications of mathematics to other sciences.