论文标题
当广义解析具有孤立的基本奇异性时
Inversion of operator pencils on Banach space using Jordan chains when the generalized resolvent has an isolated essential singularity
论文作者
论文摘要
我们假设一个有界线性算子映射一个Banach空间的广义分解在原点上具有孤立的基本奇异性，并且对以原点为中心的复合平面的某些环形区域进行了分析。在这种情况下，可以通过收敛的lourent系列在环上表示分解运算符，并且光谱集有两个组件 - - 环内边界内的有界组件和外部边界外的一个无限组件。在本文中，我们证明了域空间上的互补光谱分离投影是由相关的无限长度的载体链的各个生成子空间唯一确定的，并且域空间是这两个子空间的直接总和。我们表明，在铅笔定义的映射下生成子空间的图像为范围空间提供了相应的直接总和分解，这只是由范围空间上的互补光谱分离预测定义的分解。如果域空间具有Schauder的基础，我们表明，基本方程的分离系统被简化为两个半无限的矩阵方程系统，可以递归地求解以获得基本的解决方案，从而确定在给定环形区域上分辨率操作员的laurent系列系数。
We assume that the generalized resolvent for a bounded linear operator pencil mapping one Banach space onto another has an isolated essential singularity at the origin and is analytic on some annular region of the complex plane centred at the origin. In such cases the resolvent operator can be represented on the annulus by a convergent Laurent series and the spectral set has two components---a bounded component inside the inner boundary of the annulus and an unbounded component outside the outer boundary. In this paper we prove that the complementary spectral separation projections on the domain space are uniquely determined by the respective generating subspaces for the associated infinite-length generalized Jordan chains of vectors and that the domain space is the direct sum of these two subspaces. We show that the images of the generating subspaces under the mapping defined by the pencil provide a corresponding direct sum decomposition for the range space and that this is simply the decomposition defined by the complementary spectral separation projections on the range space. If the domain space has a Schauder basis we show that the separated systems of fundamental equations are reduced to two semi-infinite systems of matrix equations which can be solved recursively to obtain a basic solution and thereby determine the Laurent series coefficients for the resolvent operator on the given annular region.
