矩阵函数分解中的奇异积分算子的性质

Properties of Singular Integral Operators in the Matrix Functions Factorization Theory

研究奇异积分算子的性质是解决矩阵函数分解理论的重要方法和工具,但矩阵函数分解理论往往受矩阵函数类所限制。通过改进Cauchy型积分算子的作用域,提出了赫尔德函数类矩阵函数分解和对应的Toeplitz算子的基本概念,得到了换位算子的紧性结论。在此类矩阵函数分解存在的条件下,利用经典的Riemann-Hilbert问题作为工具,获得了Toeplitz算子的可逆性、核空间的维数。

Deploring the properties of singular integral operators is an important method and tool for the research on matrix function factorization theory, which is often limited by matrix function classes. The matrix function factorization of H？ lder function class and the concept of the companion Toeplitz operators are proposed based on the improvement of the Cauchy integral operatorg acting domain. The compactness of the commutator operator is proved. Using the classical RiemannHilbert problem as a tool, the reversibility of Toeplitz operator and the dimension on its kernel space are obtained.