关于Smirnov类矩阵矩阵论值函数的理论
On the Theory of Matrix Valued Functions Belonging to the Smirnov Class
摘要：
本文系统地发展了矩阵Smirnov类${\goth N}_n^+({\Bbb D})$中的矩阵值函数的矩阵论.特别地，对于Smirnov类${\goth N}_n^+({\Bbb D})$中的矩阵值函数，给出了V. I. Smirnov的最大值原理、内外分解、外函数的Smirnov-Beurling特征和Frostman定理的一个类似定理.我们还考虑了一个家庭$F_λ =F-λI$属于矩阵Smirnov类的功能是由一个复杂的参数$λ$索引。我们表明，除了一个“非常小”的集，这样的$λ$相应的内部因子的内外因式分解的功能$F_λ$是一个Blaschke-Potapov产品。本文的主要目标是为用户提供一个标准的参考源有关的矩阵Smirnov类的解析矩阵函数理论。
英文摘要：
A theory of matrix-valued functions from the matricial Smirnov class ${\goth N}_n^+({\Bbb D})$ is systematically developed. In particular, the maximum principle of V.I.Smirnov, inner-outer factorization, the Smirnov-Beurling characterization of outer functions and an analogue of Frostman's theorem are presented for matrix-valued functions from the Smirnov class ${\goth N}_n^+({\Bbb D})$. We also consider a family $F_λ =F-λI$ of functions belonging to the matricial Smirnov class which is indexed by a complex parameter $λ$. We show that with the exception of a ''very small'' set of such $λ$ the corresponding inner factor in the inner-outer factorization of the function $F_λ$ is a Blaschke-Potapov product. The main goal of this paper is to provide users of analytic matrix-function theory with a standard source for references related to the matricial Smirnov class.
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