谱系的若干结论及新应用

Some New Properties in Spectral Family and Application to Functional Analysis

谱系与分布函数的定义很相似,那么它是否具有分布函数的某些性质呢?答案是肯定的。文章主要的结果为:1.证明了对Hilbert空间中的任意一点,谱系在强收敛意义下的间断点最多可数;2.以谱系为工具证明可分的Hilbert空间中的有界自伴算子的点谱最多可数;3.用较简洁的方法证明了有界自伴算子的特征值与其谱系的关系;4.不使用Riesz-Schauder理论,以谱系为工具证明了紧自伴算子的特征值要么是有限个,要么是以0为唯一聚点的可数个,且证明过程简单;5.不使用Riesz-Schauder理论,以谱系为工具证明了紧自伴算子可以用投影算子依算子范数逼近,且证明过程简单。

Definition of spectral family and distribution functions are very similar, and they have some similar properties. The main results of this paper are in the following areas： First, there is an at most countable discontinuity point in spectral family of Hilbert space at strong convergence; Second, using spectral family to prove bound self-adjoint operators’ point spectrum is at most a countable set in separable Hilbert space; Third, simple proof of relationship about bound self-adjoint operators eigenvalue and their spectral family is given;Fourth;by using a different way from Riesz-Schauder theory,it proves distribution of compact self-adjoint operators eigenvalues; Fifth, spectral family also be applied to prove compact self-adjoint operators that can be approximated by projection operators in the sense of operator norm.