算子的扰动与谱的逼近
Perturbation of operators and approximation of spectrum
摘要：
设A（x）是可分希尔伯特空间H上的一个全纯有界自伴算子族，A（x）n是A（x）对H的一个规范正交基的前n个元的跨度的正交压缩.本文考虑的问题是用A（x）_n的特征值序列来逼近A（x）的谱。证明了A（x）的本质谱的界和在本质谱界之外的离散谱值在所有紧子集上都可以用A（x）n的特征值函数序列一致逼近.将有界自伴算子的已知结果推广到全纯算子族的情形。本文还试图预测A（0）= A的本质谱的边界之间可能出现的谱隙的存在，并研究了算子全纯微扰在谱隙预测中的作用。作为例子，讨论了某些块Toeplitz-Laurent算子的间隙问题.纯线性代数方法是这里的结果的主要优点。
英文摘要：
Let A(x) be a holomorphic family of bounded self-adjoint operators on a separable Hilbert space H and let A(x)_n be the orthogonal compressions of A(x) to the span of first n elements of an orthonormal basis of H. The problem considered here is to approximate the spectrum of A(x) using the sequence of eigenvalues of A(x)_n. We show that the bounds of the essential spectrum and the discrete spectral values outside the bounds of essential spectrum of A(x) can be approximated uniformly on all compact subsets by the sequence of eigenvalue functions of A(x)_n. The known results for a bounded selfadjoint operator, are translated into the case of a holomorphic family of operators. Also an attempt is made to predict the existence of spectral gaps that may occur between the bounds of essential spectrum of A(0) = A and study the effect of holomorphic perturbation of operators in the prediction of spectral gaps. As an example, gap issues of some block Toeplitz-Laurent operators are discussed. The pure linear algebraic approach is the main advantage of the results here.
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