Abstract

In a countably normed space of smooth functions on the unit circle, we consider the Riemann boundary value problem operator with smooth coefficients. The paper introduces the concept of smooth degenerate factorizations of types of plus and minus functions that are smooth on the unit circle. Criteria for the existence of such factorizations are given. An apparatus is proposed for calculating the indices of these factorizations in terms of coefficients. In terms of smooth degenerate factorizations, a criterion for the invertibility of the Riemann boundary value problem operator is obtained. This allows describing the spectrum of this operator. Relationships between the spectra of the Riemann operator in the spaces of smooth and summable functions with the same coefficients are indicated.

中文翻译：

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圆上光滑函数可数范空间中黎曼边值问题算子的可逆性和谱

摘要

在单位圆上光滑函数的可数赋范空间中，我们考虑具有光滑系数的黎曼边值问题算子。本文介绍了单位圆上平滑的正负函数类型的平滑简并因式分解的概念。给出了此类因式分解存在的标准。提出了一种用于根据系数计算这些因式分解的指数的装置。根据平滑简并分解，得到了黎曼边值问题算子可逆性的判据。这允许描述该算子的频谱。指出了具有相同系数的光滑函数和可求函数空间中黎曼算子的谱之间的关系。